The full basis theorem does not imply analytic wellordering

TL;DR

This paper constructs a model of ZFC where every non-empty analytically definable set of reals contains an analytically definable real, yet there is no analytically definable wellordering of the continuum, showing the full basis theorem does not imply analytic wellordering.

Contribution

It demonstrates that the full basis theorem does not necessarily lead to an analytically definable wellordering of the continuum by constructing a specific model.

Findings

Every non-empty lightface analytically definable set of reals contains an analytically definable real.

There is no analytically definable wellordering of the continuum in the constructed model.

The model uses a finite support product of Jensen minimal $oldsymbol{ m extPi^1_2}$ singleton forcing.

Abstract

We make use of a finite support product of $\omega_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the continuum.

Equations

$$T(\to 0)=T{\upharpoonright }_{\text{stem}(T){}^{⌢}0}\text{and}T(\to 1)=T{\upharpoonright }_{\text{stem}(T){}^{⌢}1}$$

$$T(\to u)=T(\to u(0))(\to u(1))(\to u(2))\dots (\to u(n-1))$$

$$\begin{array}[]{rcl}[{\boldsymbol{p}}]&=&\{\hskip 0.43057pt{x\in 2^{\omega_{1}\times\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\forall\,}\langle\xi,k\rangle\in|{\boldsymbol{p}}|\,(x(\xi,k)\in[T^{{\boldsymbol{p}}}_{\xi k}])\hskip 0.43057pt\}=\\[6.45831pt] &=&\{\hskip 0.43057pt{x\in 2^{\omega_{1}\times\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\forall\,}\langle\xi,k\rangle\in|{\boldsymbol{p}}|\;{\forall\,}m\,(x(\xi,k){\hskip 0.86108pt{\restriction}\hskip 0.86108pt}m\in T^{{\boldsymbol{p}}}_{\xi k})\hskip 0.43057pt\}\,.\end{array}$$

$${\bigcup }_{\mathbf{p}\in U_1}[\mathbf{p}]\cap \dots \cap {\bigcup }_{\mathbf{p}\in U_k}[\mathbf{p}]=[{\mathbf{u}}_1]\cup \dots \cup [{\mathbf{u}}_n],$$

$$G(\xi ,k)=\{T_{\xi k}^{\mathbf{p}}:\text{linebreak}[0]\mathbf{p}\in G\wedge ⟨\xi ,k⟩\in |\mathbf{p}|\}\subseteq \mathbf{\pi }(\xi ,k)$$

$$(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})(\xi,k)=\left\{\begin{array}[]{rcl}\boldsymbol{\pi}(\xi,k),&\text{whenever}&\langle\xi,k\rangle\in|\boldsymbol{\pi}|\smallsetminus|\text{\leavevmode{}}|\\[4.30554pt] \text{\leavevmode{}}(\xi,k),&\text{whenever}&\langle\xi,k\rangle\in|\text{\leavevmode{}}|\smallsetminus|\boldsymbol{\pi}|\\[4.30554pt] \boldsymbol{\pi}(\xi,k)\cup\text{\leavevmode{}}(\xi,k),&\text{whenever}&\langle\xi,k\rangle\in|\boldsymbol{\pi}|\cap|\text{\leavevmode{}}|\end{array}\right.$$

$${\mathbf{D}}_{\mathbf{q}}^{|\mathbf{u}|}=\{{\mathbf{u}}^{\prime }\in \text{MT}(\mathbf{\pi }):\text{linebreak}[0]|{\mathbf{u}}^{\prime }|=|\mathbf{u}|\text{ and }{\mathbf{u}}^{\prime }\cup \mathbf{q}\in \mathbf{D}\}.$$

$$X=\{⟨\xi ,k⟩\in |\mathbf{r}|:\text{linebreak}[0]T_{\xi k}^{\mathbf{r}}\in \text{MT}(\text{leavevmode})\},Y=\{⟨\xi ,k⟩\in |\mathbf{r}|:\text{linebreak}[0]T_{\xi k}^{\mathbf{r}}\in \text{MT}(\mathbf{\pi })\}.$$

$${({\mathbf{D}}_i)}_{\mathbf{q}}^{|\mathbf{u}|}=\{{\mathbf{u}}^{\prime }\in \text{MT}(\mathbf{\pi }):\text{linebreak}[0]|{\mathbf{u}}^{\prime }|=|\mathbf{u}|\text{ and }{\mathbf{u}}^{\prime }\cup \mathbf{q}\in {\mathbf{D}}_i\}.$$

$${\mathbf{D}}_{\mathbf{q}}^{|\mathbf{u}|}=\{{\mathbf{v}}^{\prime }\in \text{MT}(\mathbf{\pi }):\text{linebreak}[0]|{\mathbf{v}}^{\prime }|=|\mathbf{v}|=|\mathbf{u}|\wedge {\mathbf{v}}^{\prime }\cup \mathbf{q}\in \mathbf{D}\}.$$

$${\Delta }_{\xi kmh}=\{\mathbf{\phi }\in \text{MS}(\mathbf{\pi }):\text{linebreak}[0]h^{\mathbf{\phi }}(\xi ,k,m)\ge h\}\in {\mathfrak{M}}^{+},\text{where }h<\omega .$$

$${\mathbf{Q}}_{\xi k,m}^{\Phi }(s)={\bigcap }_{n\ge \mathrm{lh}(s)}{\bigcup }_{t\in 2^n,s\subseteq t}{\mathbf{T}}_{\xi k,m}^{\Phi }(t),$$

$$A={\bigcup }_{s\in 2^h}[T_{\xi k,m}^{\mathbf{\phi }(j)}(s)]={\bigcup }_{s\in 2^h}[{\mathbf{T}}_{\xi k,m}^{\Phi }(s)],B={\bigcup }_{t\in 2^{h^{\prime }}}[T_{\xi \ell ,n}^{\mathbf{\phi }(j)}(t)]={\bigcup }_{t\in 2^{h^{\prime }}}[{\mathbf{T}}_{\xi \ell ,n}^{\Phi }(t)]$$

$${\mathbf{D}}_{\xi k}^{\mathbf{\pi }}(\mathbf{c})=\{\mathbf{p}\in \text{MT}(\mathbf{\pi }):\text{linebreak}[0]\mathbf{p}\text{ directly forces }\mathbf{c}\notin [T_{\xi k}^{\mathbf{p}}]\}.$$

$$\mathbf{D}(\mathbf{c},Q,\mathbf{\pi })=\{\mathbf{r}\in \text{MT}(\mathbf{\pi }):\text{linebreak}[0]\mathbf{r}\text{ directly forces }\mathbf{c}\notin [Q]\},$$

$${\mathbf{D}}_{\xi k}^{\mathbf{\pi }}(\mathbf{c})=\{\mathbf{p}\in \text{MT}(\mathbf{\pi }):\text{linebreak}[0]\mathbf{p}\text{ directly forces }\mathbf{c}\notin [T_{\xi k}^{\mathbf{p}}]\}.$$

$$\mathbf{D}(\mathbf{c},Q,\mathbf{\pi })=\{\mathbf{r}\in \text{MT}(\mathbf{\pi }):\text{linebreak}[0]\mathbf{r}\text{ directly forces }\mathbf{c}\notin [Q]\}.$$

$$\begin{array}[]{lcl}\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}&=&\text{all multitrees ${\boldsymbol{p}}\in\text{MT}$ such that $|{\boldsymbol{p}}|\subseteq\omega_{1}\times m$, }\\[4.30554pt] \text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}&=&\text{all multitrees ${\boldsymbol{p}}\in\text{MT}$ with $|{\boldsymbol{p}}|\subseteq\omega_{1}\times{(\omega\smallsetminus m)}$,}\\[4.30554pt] \text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}&=&\text{all multitrees ${\boldsymbol{p}}\in\text{MT}$ such that $|{\boldsymbol{p}}|\subseteq\omega_{1}\times\{\hskip 0.43057ptm\hskip 0.43057pt\}$,}\end{array}$$

$$\begin{array}[]{lcl}\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}&=&\text{all multiforcings $\boldsymbol{\pi}\in\text{sMF}$ such that $|\boldsymbol{\pi}|\subseteq\omega_{1}\times m$, }\\[4.30554pt] \text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}&=&\text{all multiforcings $\boldsymbol{\pi}\in\text{sMF}$ with $|\boldsymbol{\pi}|\subseteq\omega_{1}\times{(\omega\smallsetminus m)}$,}\\[4.30554pt] \text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}&=&\text{all multiforcings $\boldsymbol{\pi}\in\text{sMF}$ such that $|\boldsymbol{\pi}|\subseteq\omega_{1}\times\{\hskip 0.43057ptm\hskip 0.43057pt\}$,}\end{array}$$

$$M=\{\mu <{\omega }_1:\text{linebreak}[0]B_{\mu }^n=\{X_{\alpha }^n:\text{linebreak}[0]\alpha <\mu \}\text{ for }n=0,1,2\}$$

$$\text{vv}\mathbf{\pi }\lceil \gamma \rceil {\subset }_{\{D\cap \text{MT}(\text{vv}\text{nmi}\Pi \upharpoonright \gamma )\}}\text{vv}\mathbf{\pi }\lceil \gamma ,0\rceil .$$

$$\begin{array}[]{lclcllllll}\nmi\Pi&=&\bigcup^{\text{\tt cw}}\vv{\nmi\Pi}&=&\bigcup^{\text{\tt cw}}_{\alpha<\omega_{1}}\nmi\Pi_{\alpha}&\in&\text{MF},&\\[4.30554pt] \nmi\Pi_{<\gamma}&=&\bigcup^{\text{\tt cw}}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)&=&\bigcup^{\text{\tt cw}}_{\alpha<\gamma}\nmi\Pi_{\alpha}&\in&\text{sMF},&\text{for each $\gamma<\omega_{1}$}\\[4.30554pt] \nmi\Pi_{\geq\gamma}&=&\bigcup^{\text{\tt cw}}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}(\omega_{1}\smallsetminus\gamma))&=&\bigcup^{\text{\tt cw}}_{\gamma\leq\alpha<\omega_{1}}\nmi\Pi_{\alpha}&\in&\text{MF},&\text{for each $\gamma<\omega_{1}$}.\end{array}$$

$${\mathbb{P}}_{<\gamma }=\text{MT}(\text{nmi}{\Pi }_{<\gamma })=\text{MT}(\text{vv}\text{nmi}\Pi \upharpoonright \gamma ),{\mathbb{P}}_{\ge \gamma }=\text{MT}(\text{nmi}{\Pi }_{\ge \gamma })=\text{MT}(\text{vv}\text{nmi}\Pi \upharpoonright ({\omega }_1\setminus \gamma )).$$

$$\begin{array}[]{cclc}\alpha<\alpha(\xi,k)&\mathbin{\,\Longleftrightarrow\,}&{\exists\,}\boldsymbol{\pi}(\boldsymbol{\pi}=\nmi\Pi_{\alpha}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptk}\land\langle\xi,k\rangle\in\mathop{\tt dom}\boldsymbol{\pi})&\mathbin{\,\Longleftrightarrow\,}\\[4.30554pt] &\mathbin{\,\Longleftrightarrow\,}&{\forall\,}\boldsymbol{\pi}(\boldsymbol{\pi}=\nmi\Pi_{\alpha}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptk}\mathbin{\,\Longrightarrow\,}\langle\xi,k\rangle\in\mathop{\tt dom}\boldsymbol{\pi})&.\end{array}$$

$$C=\{\gamma <{\omega }_1:\text{linebreak}[0]A\cap {\mathbb{P}}_{<\gamma }\text{ is a maximal antichain in }{\mathbb{P}}_{<\gamma }\}$$

$$G{\upharpoonright \upharpoonright }_{<m}=G\cap \text{MT}{\upharpoonright \upharpoonright }_{<m}=\{\mathbf{p}{\upharpoonright \upharpoonright }_{<m}:\text{linebreak}[0]\mathbf{p}\in G\},$$

$$\mathbb{P}{\upharpoonright \upharpoonright }_{<m}=\mathbb{P}\cap \text{MT}{\upharpoonright \upharpoonright }_{<m}=\{\mathbf{p}{\upharpoonright \upharpoonright }_{<m}:\text{linebreak}[0]\mathbf{p}\in \mathbb{P}\}.$$

$${\mathbf{D}}_{\xi k}^{\text{nmi}\Pi }(\mathbf{c})=\{\mathbf{p}\in \mathbb{P}=\text{MT}(\text{nmi}\Pi ):\text{linebreak}[0]\mathbf{p}\text{ directly forces }\mathbf{c}\notin [T_{\xi k}^{\mathbf{p}}]\}.$$

$${\mathbb{W}}_k=\{⟨\xi ,x_{\xi k}[G]⟩:\text{linebreak}[0]\xi <{\omega }_1\}\subseteq {\omega }_1\times 2^{\omega }$$

$${\forall\,}\alpha<\omega_{1}\>{\exists\,}T\in\nmi\Pi_{\alpha}(\xi,k)\big{(}\alpha(\xi,k)\leq\alpha\mathbin{\,\Longrightarrow\,}x\in[T]\big{)},$$

Full text

1. 1 Introduction

2. 2 Comments

3. 3 The structure of the paper

4. I Basic constructions

5. 1 Perfect trees

6. 2 Perfect tree forcing notions

7. 3 Splitting construction

8. 4 Multiforcings and multitrees

9. 5 Multisystems

10. II Refinements

11. 1 Refining perfect tree forcings

12. 2 Refining multiforcings

13. 3 Generic refinement of a multiforcing

14. 4 Preservation of density

15. III Structure of real names

16. 1 Real names

17. 2 Direct forcing

18. 3 Locking real names

19. 4 Non-principal names and avoiding refinements

20. 5 Generic refinements avoid non-principal names

21. 6 Consequences for reals in generic extensions

22. 7 Combining refinement types

23. IV The forcing notion

24. 1 Increasing sequences of small multiforcings

25. 2 Layer restrictions of multiforcings and deciding sets

26. 3 Auxiliary diamond sequences

27. 4 Key sequence theorem

28. 5 Key product forcing

29. V Auxiliary forcing relation

30. 1 Auxiliary forcing: preliminaries

31. 2 Auxiliary forcing

32. 3 Forcing simple formulas

33. 4 Tail invariance

34. 5 Permutations

35. 6 Forcing with subsequences of the key multisequence

36. VI The model

37. 1 Key generic extension and subextensions

38. 2 Definability of generic reals

39. 3 Elementary equivalence

40. 4 Non-wellorderability

41. 5 Proof of the main theorem

42. References

1 Introduction

The uniformization problem, introduced by Luzin [22, 23], as well as the related basis problem, are well known in modern set theory. (See Moschovakis [24], Kechris [21], Hauser and Schindler [7] for both older and more recent studies.) In particular, it is known that every non-empty $\varSigma^{1}_{2}$ set of reals contains a $\varDelta^{1}_{2}$ real, but on the other hand, it is consistent that there exists a non-empty $\varPi^{1}_{2}$ set of reals containing even no ordinal-definable real.

The negative part of this result was strengthened in [19] to the effect that the counter-example set $X\subseteq\omega^{\omega}$ is a $\varPi^{1}_{2}$ ${\mathrel{{\text{\sf E}}_{0}}}$-equivalence class (hence, a countable set), see related discussions at the Mathoverflow exchange desk 111  A question about ordinal definable real numbers. Mathoverflow, March 09, 2010. http://mathoverflow.net/questions/17608.

and at FOM 222 Ali Enayat. Ordinal definable numbers. FOM Jul 23, 2010. http://cs.nyu.edu/pipermail/fom/2010-July/014944.html. Recall that $\mathrel{{\text{\sf E}}_{0}}$ is an equivalence relation on $\omega^{\omega}$ defined so that $x\mathrel{{\text{\sf E}}_{0}}y$ iff $x(n)=y(n)$ for all but finite $n$.

As for the positive direction, the most transparent way to get a basis result is to make use of an analytically definable wellordering $<$ of the reals, which enables one to pick the ${<}$-least real in each non-empty set of reals. This leads to the question: is the existence of an analytically definable wellordering $<$ of the reals independent of the basis theorem. We answer it in the positive:

Theorem 1.1**.**

In a suitable generic extension of ${\mathbf{L}}$, it is true that in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no lightface analytically definable wellordering of the continuum.

More precisely, there is a cardinal-preserving generic extension ${\mathbf{L}}[X]$ of ${\mathbf{L}}$, such that $X=\langle x_{\xi k}\rangle_{\xi<\omega_{1}^{\mathbf{L}}\land k<\omega}$, where each $x_{\xi k}$ is a real in $2^{\omega}$, and in addition

1. (I)

if $m<\omega$ then the submodel ${\mathbf{L}}[X_{m}]$ admits a $\varDelta^{1}_{m+3}$ wellordering of the reals of length $\omega_{1}$, where $X_{m}=\langle x_{\xi k}\rangle_{\xi<\omega_{1}^{\mathbf{L}}\land k<m}\;;$ 2. (II)

if $m<\omega$ then $\omega^{\omega}\cap{\mathbf{L}}[X_{m}]$ is a $\varSigma^{1}_{m+3}$ set in ${\mathbf{L}}[x]\;;$ 3. (III)

if $m<\omega$ then ${\mathbf{L}}[X_{m}]$ is an elementary submodel of ${\mathbf{L}}[x]$ w. r. t. all $\varSigma^{1}_{m+2}$ formulas with reals in ${\mathbf{L}}[X_{m}]$ as parameters*;*** 4. (IV)

it is true in ${\mathbf{L}}[X]$ that there is no lightface analytically definable wellordering of the reals*.***

To see that the additional claims imply the main claim (the full basis theorem), let, in ${\mathbf{L}}[X]$, $Z\subseteq\omega^{\omega}$ be a non-empty $\varSigma^{1}_{m+2}$ set of reals. Then $Z^{\prime}=Z\cap{\mathbf{L}}[X_{m}]$ is a $\varSigma^{1}_{m+3}$ set by (II), and $Z^{\prime}\neq\varnothing$ by (III). It remains to pick the least real in $Z^{\prime}$ in the sense of the lightface $\varDelta^{1}_{m+3}$ wellordering given by (I).

2 Comments

To prove the theorem, we define, in ${\mathbf{L}}$, a system of forcing notions ${{\mathbb{P}}}_{\xi k}$, $\xi<\omega_{1}$ and $k<\omega$, whose finite-support product ${{\mathbb{P}}}=\prod_{\xi,k}{{\mathbb{P}}}_{\xi k}$ adds an array $X=\langle x_{\xi k}\rangle_{\xi<\omega_{1},k<\omega}$ of reals $x_{\xi k}$ to ${\mathbf{L}}$, such that (I), (II), (III), (IV) hold in ${\mathbf{L}}[X]$.

Regarding the history of this research, in goes down to Jensen [10], where a forcing ${{\mathbb{J}}}=\bigcup_{\alpha<\omega_{1}}{{\mathbb{J}}}_{\alpha}$ is defined in ${\mathbf{L}}$, the constructible universe, such that each ${{\mathbb{J}}}_{\alpha}$ is a countable set of perfect trees in $2^{<\omega},$ the canonical ${{{\mathbb{J}}}}$-generic real is a single ${{{\mathbb{J}}}}$-generic real in the extension, and ‘being a ${{{\mathbb{J}}}}$-generic real’ is a $\varPi^{1}_{2}$ property, so as a result we get a $\varPi^{1}_{2}$ nonconstructible singleton in any ${{{\mathbb{J}}}}$-generic extension of ${\mathbf{L}}$. See 28A in [8] for a more modern exposition of Jensen’s forcing.

A nonconstructible $\varPi^{1}_{2}$ singleton also was defined in [9] by means of the almost-disjoint forcing, yet the construction in [10] has the advantage of minimality of ${{{\mathbb{J}}}}$-generic reals and some other advantages (as well as some disadvantages).

Jensen’s forcing construction (including its iterations) was exploited by Abraham [1, 2], including a definable minimal collapsing real. Another modification of Jensen’s forcing construction in [11] yields such a forcing notion in ${\mathbf{L}}$ that any extension of ${\mathbf{L}}$, containing two generic reals $x\neq y$, necessarily satisfies $\omega_{1}^{\mathbf{L}}<\omega_{1}$. See [3, 15] on some other modifications in coding purposes.

A different modification of Jensen’s forcing construction was engineered in [16] in order to define an extension of ${\mathbf{L}}$ in which, for a given $n\geq 2$, there is a nonconstructible $\varPi^{1}_{n}$ singleton while all $\varSigma^{1}_{2}$ reals are constructible. (An abstract appeared in [14].) The idea is to complicate the inductive construction of Jensen’s sequence $\vec{{\mathbb{J}}}=\langle{{\mathbb{J}}}_{\alpha}\rangle_{\alpha<\omega_{1}}$ in ${\mathbf{L}}$ by the requirement that it intersects any set, of a certain definability level, dense in the collection of all possible countable initial steps of the construction. The same inner genericity idea, with respect to the Jensen – Johnsbraten forcing notion in [11], was developed in [18].

Such an inner genericity modification of the Jensen – Solovay almost-disjoint forcing [9] was developed in [6] towards some great results which unfortunately have never been published in a mathematical journal. Except for a one result, a model in which the set of all analytically definable reals is equal to the set of all constructible reals, independently obtained in [17]. We employ the inner definable genericity idea here in such a way that if $m<\omega$ then the ${m}$-tail $\langle{{\mathbb{P}}}_{\xi k}\rangle_{\xi<\omega_{1}\land k\geq m}$ of the forcing construction, bears an amount of inner definable genericity which strictly depends on $m$. (See Definition 2.1, where a key concept is introduced.)

Ali Enayat (Footnote 2) conjectured that some definability questions can be solved by finite-support products of Jensen’s [10] forcing ${{\mathbb{J}}}$. Enayat demonstrated in [4] that a symmetric part of the ${{{\mathbb{J}}}^{\omega}}$-generic extension of ${\mathbf{L}}$ definitely yields a model of ZF (not a model of ZFC!) in which there is a Dedekind-finite infinite $\varPi^{1}_{2}$ set of reals with no OD elements. Following the conjecture, we proved in [12] that indeed it is true in a ${{{\mathbb{J}}}^{\omega}}$-generic extension of ${\mathbf{L}}$ that the set of ${{{\mathbb{J}}}}$-generic reals is a countable non-empty $\varPi^{1}_{2}$ set with no OD elements. We also proved in [13] that the existence of a $\varPi^{1}_{2}$ ${\mathrel{{\text{\sf E}}_{0}}}$-class with no OD elements is consistent with ZFC, using a ${\mathrel{{\text{\sf E}}_{0}}}$-invariant version of Jensen’s forcing. We further employed another finite-support product of Jensen’s forcing to define a generic extension of ${\mathbf{L}}$ where there is a $\varPi^{1}_{2}$ set $P\subseteq\omega^{\omega}\times\omega^{\omega}$ which has countable cross-sections $P_{x}=\{\hskip 0.43057pt{y\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\langle x,y\rangle\in P\hskip 0.43057pt\}$ and is non-uniformizable by any projective set [20].

Acknowledgement**.**

The idea of making use of a suitable finite-support product of Jensen-like forcing notions in order to obtain a model, in which the full basis theorem holds but there is no lightface analytically definable wellordering of the continuum, was communicated to an author of this paper (VK) by Ali Enayat in 2015, and we thank Ali Enayat for fruitful discussions and helpful ideas.

3 The structure of the paper

The general organization of the paper is as follows. Chapter I contains a general formalism related to forcing by perfect trees and finite-support products, convenient for our goals. Following Jensen [10], we consider forcing notions of the form ${{\mathbb{P}}}=\bigcup_{\alpha<\lambda}{{\mathbb{P}}}_{\alpha}$, where $\lambda<\omega_{1}$ and each ${{\mathbb{P}}}_{\alpha}$ is a countable set of perfect trees in $2^{<\omega}.$ Each term ${{\mathbb{P}}}_{\alpha}$ has to satisfy some routine conditions of refinement with respect to the previous terms, in particular, to make sure that each ${{\mathbb{P}}}_{\alpha}$ remains pre-dense at further steps. Also, each ${{\mathbb{P}}}_{\alpha}$ has to lock some dense sets in $\bigcup_{\xi<\alpha}{{\mathbb{P}}}_{\xi}$ so that they remain pre-dense at further steps as well. And this procedure has to be extended from single forcing notions to their finite-support products. These issues are dealt with in Chapter II.

Then we consider real names with respect to finite-support products of perfect-tree forcing notions in Chapter III. Here the key issue is to make sure that if ${{\mathbb{P}}}$ is a factor in a product forcing considered then there is no other ${{{\mathbb{P}}}}$-generic real in the whole product extension except for the obvious one.

In Chapter IV we define the forcing notion $\boldsymbol{\mathbb{P}}=\prod_{\xi<\omega_{1},k<\omega}{{\mathbb{P}}}_{\xi k}$ to prove the main theorem, in the form of a limit of a certain increasing sequence of countable products of countable perfect-tree forcing notions. Quite a complicated construction of this sequence in ${\mathbf{L}}$ involves ideas related to diamond-style constructions, as well as to some sort of definable genericity, as explained above.

The forcing ${{\mathbb{P}}}$ is not analytically definable; basically, each ${k}$-th layer $\langle{{\mathbb{P}}}_{\xi k}\rangle_{\xi<\omega_{1}}$ belongs to $\varDelta^{1}_{k+4}$. But it is a key property that the ${\boldsymbol{\mathbb{P}}}$-forcing relation restricted to $\varSigma^{1}_{n}$ formulas is essentially $\varSigma^{1}_{n}$. We prove this in Chapter V, with the help of an auxiliary forcing notion $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$. We also establish the invariance of $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$ with respect to countable-support permutations of $\omega_{1}\times\omega$.

We finally prove Theorem 1.1 in Chapter VI, on the base of the results obtained in two previous chapters.

I Basic constructions

We begin with some basic things: perfect trees in the Cantor space $2^{\omega}$, perfect tree forcing notions (those which consist of perfect trees), their finite-support products, and a splitting construction of perfect trees.

1 Perfect trees

Let $2^{<\omega}$ be the set of all strings (finite sequences) of numbers $0,1$. If $t\in 2^{<\omega}$ and $i=0,1$ then $t{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}k$ is the extension of $t$ by $k$. If $s,t\in 2^{<\omega}$ then $s\subseteq t$ means that $t$ extends $s$, while $s\subset t$ means proper extension. If $s\in 2^{<\omega}$ then $\mathop{\tt lh}(s)$ is the length of $s$, and $2^{n}=\{\hskip 0.43057pt{s\in 2^{<\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\mathop{\tt lh}(s)=n\hskip 0.43057pt\}$ (strings of length $n$).

A set $T\subseteq 2^{<\omega}$ is a tree iff for any strings $s\subset t$ in $2^{<\omega}$, if $t\in T$ then $s\in T$. Every non-empty tree $T\subseteq 2^{<\omega}$ contains the empty string $\Lambda$. If $T\subseteq 2^{<\omega}$ is a tree and $s\in T$ then put $T{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}=\{\hskip 0.43057pt{t\in T\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\subseteq t\lor t\subseteq s\hskip 0.43057pt\}$.

Let PT be the set of all perfect trees $\varnothing\neq T\subseteq 2^{<\omega}$. Thus a non-empty tree $T\subseteq 2^{<\omega}$ belongs to PT iff it has no endpoints and no isolated branches. Then there is a largest string $s\in T$ such that $T=T{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}$; it is denoted by $s=\text{\tt stem}(T)$ (the stem of $T$); we have $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}1\in T$ and $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}0\in T$ in this case.

Definition 1.1** (perfect sets).**

If $T\in\text{PT}$ then $[T]=\{\hskip 0.43057pt{a\in 2^{\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\forall\,}n\,(a{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}n\in T)\hskip 0.43057pt\}$ is the set of all paths through $T$, a perfect set in $2^{\omega}$. Conversely if $X\subseteq 2^{\omega}$ is a perfect set then $\mathop{\tt tree}(X)=\{\hskip 0.43057pt{a{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}n\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pta\in X\land n<\omega\hskip 0.43057pt\}\in\text{PT}$ and $[\mathop{\tt tree}(X)]=X$.

Trees $T,S\in\text{PT}$ are almost disjoint, a. d. for brevity, iff

the intersection $S\cap T$ is finite; this is equivalent to just $[S]\cap[T]=\varnothing$. ∎

The simple splitting of a tree $T\in\text{PT}$ consists of smaller trees

[TABLE]

in PT, so that $[T{(\to i)}]=\{\hskip 0.43057pt{x\in[T]\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptx(h)=i\hskip 0.43057pt\}$, where $h=\mathop{\tt lh}(\text{\tt stem}(T))$. We let

[TABLE]

for each string $u\in 2^{<\omega}\hskip 0.86108pt,\linebreak[0]\>\mathop{\tt lh}(u)=n$; and separately $T{(\to\Lambda)}=T$.

Lemma 1.2**.**

Suppose that $T\in\text{PT}$. Then$:$

1. (i)

if $u\in 2^{<\omega}$ then there is a string $s\in 2^{<\omega}$ such that $T{(\to u)}={T}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}\;;$ 2. (ii)

if $s\in 2^{<\omega}$ then there is a string $u\in 2^{<\omega}$ such that ${T}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}=T{(\to u)}\;;$ 3. (iii)

if $\varnothing\neq U\subseteq[T]$ is a (relatively) open subset of $[T]$, or at least $U$ has a non-empty interior in $[T]$, then there is a string $s\in T$ such that ${T}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}\subseteq U$. ∎

If $T\in\text{PT}$ and $a\in 2^{\omega}$ then the intersection $T{(\to a)}=\bigcap_{n<\omega}T{(\to a{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}n)}=\{\hskip 0.43057pt\boldsymbol{h}_{T}(a)\hskip 0.43057pt\}$ is a singleton, and the map $\boldsymbol{h}_{T}$ is a canonical homeomorphism from $2^{\omega}$ onto $[T]$. Accordingly if $S,T\in\text{PT}$ then the map $\boldsymbol{h}_{ST}(x)=\boldsymbol{h}_{T}({\boldsymbol{h}_{S}}^{-1}(x))$ is a canonical homeomorphism from $[S]$ onto $[T]$.

2 Perfect tree forcing notions

A perfect-tree forcing notion is any non-empty set

${{\mathbb{P}}}\subseteq\text{PT}$ such that if $s\in T\in{{\mathbb{P}}}$ then $T{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}\in{{\mathbb{P}}}$, or equivalently, by Lemma 1.2, if $u\in 2^{<\omega}$ then $T{(\to u)}\in{{\mathbb{P}}}$. Let PTF be the set of all such forcing notions

${{\mathbb{P}}}\subseteq\text{PT}$.

Example 2.1**.**

If $s\in 2^{<\omega}$ then the tree $[s]=\{\hskip 0.43057pt{t\in 2^{<\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\subseteq t\lor t\subseteq s\hskip 0.43057pt\}$ belongs to PT. The set ${{\mathbb{P}}}_{\text{\rm coh}}=\{\hskip 0.43057pt{[s]\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\in 2^{<\omega}\hskip 0.43057pt\}$ of all such trees (the Cohen forcing)

is a regular perfect-tree forcing notion. ∎

Lemma 2.2**.**

Let ${{\mathbb{P}}}\in\text{{PTF}}$. If $T\in{{\mathbb{P}}}$ and a set $X\subseteq[T]$ is (relatively) open (resp., clopen) in $[T]$, then there is a countable (resp., finite) set $\mathscr{S}$ of pairwise a. d. trees $S\in{{\mathbb{P}}}$, satisfying $\bigcup_{S\in\mathscr{S}}[S]=X$.∎

Lemma 2.3**.**

1. (i)

If $s\in T\in{{\mathbb{P}}}\in\text{{PTF}}$ then $T{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}\in{{\mathbb{P}}}$. 2. (ii)

If ${{\mathbb{P}}},{{\mathbb{P}}}^{\prime}\in\text{{PTF}}$, $T\in{{\mathbb{P}}}$, $T^{\prime}\in{{\mathbb{P}}}^{\prime}$, then there are trees $S\in{{\mathbb{P}}}$, $S^{\prime}\in{{\mathbb{P}}}^{\prime}$ such that $S\subseteq T$, $S^{\prime}\subseteq T^{\prime}$, and $[S]\cap[S^{\prime}]=\varnothing$.

Proof**.**

(i) use Lemma 1.2. (ii) If $T=T^{\prime}$ then let $S=T{(\to 0)}$, $S^{\prime}=T{(\to 1)}$. If say $T\not\subseteq T^{\prime}$ then let $s\in T\smallsetminus T^{\prime}$, $S=T{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}$, and simply $S^{\prime}=T^{\prime}$. ∎

Definition 2.4**.**

A set ${{\mathbb{A}}}\subseteq\text{PT}$ is an antichain iff any trees

$T\neq T^{\prime}$ in ${{\mathbb{A}}}$ are a. d., that is, $[T]\cap[T^{\prime}]=\varnothing$. A forcing notion ${{\mathbb{P}}}\in\text{{PTF}}$ is:

small,

if it is countable;

special,

if there is an antichain ${{\mathbb{A}}}\subseteq{{\mathbb{P}}}$ such that ${{\mathbb{P}}}=\{\hskip 0.43057pt{T{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\in T\in{{\mathbb{A}}}\hskip 0.43057pt\}$ — note that ${{\mathbb{A}}}$ is unique if exists; we write ${{\mathbb{A}}}=\mathop{\tt base}({{\mathbb{P}}})$ (the base of ${{\mathbb{P}}}$);

regular,

if for any $S,T\in{{\mathbb{P}}}$, the intersection $[S]\cap[T]$ is clopen in $[S]$ or clopen in $[T]$ (or clopen in both $[S]$ and $[T]$). ∎

Lemma 2.5**.**

Let ${{\mathbb{P}}}\in\text{{PTF}}$. If ${{\mathbb{P}}}$ is special and $S,T\in{{\mathbb{P}}}$ are not a. d., then they are comparable$:$ $S\subseteq T$ or $T\subseteq S$.

If ${{\mathbb{P}}}$ is special then ${{\mathbb{P}}}$ is regular. If ${{\mathbb{P}}}$ is regular, then

1. (i)

if $S,T\in{{\mathbb{P}}}$ are not a. d., then they are compatible in ${{\mathbb{P}}}$, that is, there is a tree $R\in{{\mathbb{P}}}$ such that $R\subseteq S\cap T$. 2. (ii)

if $S_{1},\dots,S_{k}\in{{\mathbb{P}}}$ then there is a finite set of pairwise a. d. trees $R_{1},\dots,R_{n}\in{{\mathbb{P}}}$ such that $[S_{1}]\cap\ldots\cap[S_{k}]=[R_{1}]\cup\ldots\cup[R_{n}]$. 3. (iii)

if $\mathscr{S}_{1},\dots,\mathscr{S}_{k}$ are finite collections of trees in ${{\mathbb{P}}}$ then there is a finite set of trees $R_{1},\dots,R_{n}\in{{\mathbb{P}}}$ such that ${\bigcup_{S\in\mathscr{S}_{1}}[S]}\cap\ldots\cap{\bigcup_{S\in\mathscr{S}_{k}}[S]}=[R_{1}]\cup\ldots\cup[R_{n}]$, and for any $\mathscr{S}_{i}$ and $R_{j}$, there is $S\in\mathscr{S}_{i}$ such that $R_{j}\subseteq S$.

Proof**.**

(iii) Apply (ii) to every set of the form $[S_{1}]\cap\ldots\cap[S_{k}]$, where $S_{i}\in\mathscr{S}_{i}$, ${\forall\,}i$, then gather all trees $R_{i}$ obtained in one finite set. ∎

Remark 2.6**.**

Any set ${{\mathbb{P}}}\in\text{{PTF}}$ can be considered as a forcing notion (if $T\subseteq T^{\prime}$ then $T$ is a stronger condition); then ${{\mathbb{P}}}$ adds a real $x\in 2^{\omega}$. ∎

Lemma 2.7**.**

If a set $G\subseteq{{\mathbb{P}}}$ is generic over a ground set universe ${\mathbf{V}}$ (resp., over a transitive model, e. g. ${\mathbf{L}}$) then

1. (i)

the intersection $\bigcap_{T\in G}[T]$ contains a single real $x=x[G]\in 2^{\omega}$, and 2. (ii)

*this real $x$ is ${{{\mathbb{P}}}}$-generic, *in the sense that if $D\subseteq{{\mathbb{P}}}$ is dense in ${{\mathbb{P}}}$ and belongs to ${\mathbf{V}}$ (resp., to the ground model) then $x\in\bigcup_{T\in D}[T]$.∎

As usual, a set $D\subseteq{{\mathbb{P}}}$ is:

• $-$

open in ${{\mathbb{P}}}$, if for any trees $T\subseteq S$ in ${{\mathbb{P}}}$, $T\in D\mathbin{\,\Longrightarrow\,}S\in D$;

• $-$

dense in ${{\mathbb{P}}}$, if for any $T\in{{\mathbb{P}}}$ there is $S\in D$, $S\subseteq T$;

• $-$

pre-dense in ${{\mathbb{P}}}$, if the set $D^{\prime}=\{\hskip 0.43057pt{T\in{{\mathbb{P}}}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\exists\,}S\in D\,(T\subseteq S)\hskip 0.43057pt\}$ is dense in ${{\mathbb{P}}}$.

3 Splitting construction

We proceed with an important splitting/fusion construction of perfect trees by means of infinite splitting systems of such trees.

Definition 3.1**.**

Let FSS be the set of all finite splitting systems, that is, systems of the form $\varphi=\langle T_{s}\rangle_{s\in 2^{\leq n}}$, where $n=\text{\tt hgt}(\varphi)<\omega$ (the height of $\varphi$), each value $T_{s}=T^{\varphi}_{s}=\varphi(s)$ is a tree in PT, and

1. $(*)$

if $s\in 2^{<n}$ and $i=0,1$ (so $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i\in 2^{\leq n}$) then $T_{s{\mathbin{\hskip 0.60275pt{}^{\smallfrown}}}i}\subseteq T_{s}{(\to i)}$ — it easily follows that $[T_{s{\mathbin{\hskip 0.60275pt{}^{\smallfrown}}}0}]\cap[T_{s{\mathbin{\hskip 0.60275pt{}^{\smallfrown}}}1}]=\varnothing$.

We add the empty system $\boldsymbol{\Lambda}$ to FSS, with $\text{\tt hgt}(\boldsymbol{\Lambda})=-1$. ∎

A tree $T$ occurs in $\varphi\in\text{\bf FSS}$ if $T=\varphi(s)$ for some $s\in 2^{\leq\text{\tt hgt}(\varphi)}$. If all trees occurring in $\varphi$ belong to some ${{\mathbb{P}}}\in\text{{PTF}}$ then say that $\varphi$ is a finite splitting system over ${{\mathbb{P}}}$, symbolically $\varphi\in\text{\bf FSS}({{\mathbb{P}}})$.

Let $\varphi,\psi$ be systems in FSS. Say that $\varphi$ extends $\psi$, symbolically $\psi\preccurlyeq\varphi$, if $n=\text{\tt hgt}(\psi)\leq\text{\tt hgt}(\varphi)$ and $\psi(s)=\varphi(s)$ for all $s\in 2^{\leq n}$, and properly extends, $\psi\prec\varphi$, if in fact $\text{\tt hgt}(\psi)<\text{\tt hgt}(\varphi)$ strictly.

Each system $\varphi\in\text{\bf FSS}({{\mathbb{P}}})$ with $\text{\tt hgt}(\varphi)=0$ consists essentially of a single tree $T^{\varphi}_{\Lambda}\in{{\mathbb{P}}}$. The next lemma provides systems of arbitrary height.

Lemma 3.2**.**

Assume that ${{\mathbb{P}}}\in\text{{PTF}}$. If $n\geq 1$ and $\psi=\langle T_{s}\rangle_{s\in 2^{\leq n}}\in\text{\bf FSS}({{\mathbb{P}}})$ then there is a system $\varphi=\langle T_{s}\rangle_{s\in 2^{\leq n+1}}\in\text{\bf FSS}({{\mathbb{P}}})$ which properly extends $\psi$.

Proof**.**

If $s\in 2^{n}$ and $i=0,1$ then let $T_{s{\mathbin{\hskip 0.60275pt{}^{\smallfrown}}}i}=T_{s}{(\to i)}$. ∎

The next well-known lemma belongs to the type of splitting/fusion lemmas widely used in connection with the perfect set forcing and some similar forcings.

Lemma 3.3**.**

Let ${{\mathbb{P}}}\in\text{{PTF}}$. Then there is an ${\prec}$-increasing sequence $\langle\varphi_{n}\rangle_{n<\omega}$ of systems in $\text{\bf FSS}({{\mathbb{P}}})$. And if $\langle\varphi_{n}\rangle_{n<\omega}$ is such then$:$

1. (i)

the limit system $\varphi=\bigcup_{n}\varphi_{n}=\langle T_{s}\rangle_{s\in 2^{<\omega}}$ satisfies $(*)$ ‣ 3.1 of Definition 3.1 on the whole domain of strings $s\in 2^{<\omega}\,;$ 2. (ii)

$T=\bigcap_{n}\bigcup_{s\in 2^{n}}T_{s}$* is a perfect tree in PT and $[T]=\bigcap_{n}\bigcup_{s\in 2^{n}}[T_{s}]\,;$* 3. (iii)

if $u\in 2^{<\omega}$ then $T{(\to u)}=T\cap T_{u}=\bigcap_{n\geq\mathop{\tt lh}(u)}\bigcup_{s\in 2^{n},u\subseteq s}T_{s}$. ∎

4 Multiforcings and multitrees

We’ll systematically make use of finite support products of perfect tree forcings in this paper. The following definitions introduce suitable notation.

Call a multiforcing

any map $\boldsymbol{\pi}:|\boldsymbol{\pi}|\to\text{{PTF}}$, where $|\boldsymbol{\pi}|=\mathop{\tt dom}\boldsymbol{\pi}\subseteq\omega_{1}\times\omega$. Thus each set ${\boldsymbol{\pi}}(\xi,k)$,

$\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, is a perfect tree forcing notion. Such a $\boldsymbol{\pi}$ is:

• $-$

small, if both $|\boldsymbol{\pi}|$ and each forcing ${\boldsymbol{\pi}}(\xi,k)$, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, are countable;

• $-$

special, if each ${\boldsymbol{\pi}}(\xi,k)$ is special in the sense of Definition 2.4;

• $-$

regular, if each ${\boldsymbol{\pi}}(\xi,k)$ is regular, in the sense of Definition 2.4.

Let MF be the set of all multiforcings.

Let a multitree be any map ${\boldsymbol{p}}:|{\boldsymbol{p}}|\to\text{PT}$, such that $|{\boldsymbol{p}}|=\mathop{\tt dom}{\boldsymbol{p}}\subseteq\omega_{1}\times\omega$ is finite and each value $T^{{\boldsymbol{p}}}_{\xi k}={\boldsymbol{p}}(\xi,k)$ is a tree in PT.

In this case we define a cofinite-dimensional perfect cube in $2^{\omega_{1}\times\omega}$

[TABLE]

Let MT be the set of all multitrees.

We order MT componentwise: ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$ (${\boldsymbol{q}}$ is stronger) iff $|{\boldsymbol{p}}|\subseteq|{\boldsymbol{q}}|$ and $T^{{\boldsymbol{q}}}_{\xi k}\subseteq T^{{\boldsymbol{p}}}_{\xi k}$ for all $\langle\xi,k\rangle\in|{\boldsymbol{p}}|$; this is equivalent to $[{\boldsymbol{q}}]\subseteq[{\boldsymbol{p}}]$, so that stronger multitrees correspond to smaller cubes. The weakest multitree $\boldsymbol{\Lambda}\in\text{MT}$ is just the empty map;

$|\boldsymbol{\Lambda}|=\varnothing$ and $[\boldsymbol{\Lambda}]=2^{\omega_{1}\times\omega}$.

Multitrees ${\boldsymbol{p}},{\boldsymbol{q}}$ are somewhere almost disjoint, or s. a. d., if, for at least one pair of indices

$\langle\xi,k\rangle\in|{\boldsymbol{p}}|\cap|{\boldsymbol{q}}|$, the trees $T^{{\boldsymbol{p}}}_{\xi k}$, $T^{{\boldsymbol{q}}}_{\xi k}$ are a. d., that is, $[T^{{\boldsymbol{p}}}_{\xi k}]\cap[T^{{\boldsymbol{q}}}_{\xi k}]=\varnothing$, or equivalently, $T^{{\boldsymbol{p}}}_{\xi k}\cap T^{{\boldsymbol{q}}}_{\xi k}$ is finite.

Corollary 4.1** (of Lemma 2.5(i)).**

If $\boldsymbol{\pi}$ is a regular multiforcing and multitrees ${\boldsymbol{p}},{\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ are not s. a. d., then ${\boldsymbol{p}},{\boldsymbol{q}}$ are compatible in $\text{MT}(\boldsymbol{\pi})$, so that there is a multitree ${\boldsymbol{r}}\in\text{MT}(\boldsymbol{\pi})$ with ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}$, ${\boldsymbol{r}}\leqslant{\boldsymbol{q}}$.∎

If $\boldsymbol{\pi}$ is a multiforcing then a ${\boldsymbol{\pi}}$-multitree is any multitree ${\boldsymbol{p}}$ with $|{\boldsymbol{p}}|\subseteq|\boldsymbol{\pi}|$ and $T^{{\boldsymbol{p}}}_{\xi k}\in{\boldsymbol{\pi}}(\xi,k)$ for all $\langle\xi,k\rangle\in|{\boldsymbol{p}}|$. Let $\text{MT}(\boldsymbol{\pi})$

be the set of all ${\boldsymbol{\pi}}$-multitrees; it is equal to the finite support product $\prod_{\langle\xi,k\rangle\in|\boldsymbol{\pi}|}{\boldsymbol{\pi}}(\xi,k)$.

The following is similar to Lemma 2.5(iii).

Lemma 4.2**.**

If a multiforcing $\boldsymbol{\pi}$ is regular, $\xi\subseteq|\boldsymbol{\pi}|$ is finite, and $U_{1},\dots,U_{k}$ are finite collections of multitrees in $\text{MT}(\boldsymbol{\pi})$ with $|{\boldsymbol{p}}|=\xi$ for all ${\boldsymbol{p}}\in\bigcup_{i}U_{i}$, then there is a finite set of multitrees $\boldsymbol{u}_{1},\dots,\boldsymbol{u}_{n}\in\text{MT}(\boldsymbol{\pi})$ such that $|\boldsymbol{u}_{j}|=\xi$, ${\forall\,}j$,

[TABLE]

and for any $U_{i}$ and $\boldsymbol{u}_{j}$, there is ${\boldsymbol{p}}\in U_{i}$ such that $[\boldsymbol{u}_{j}]\subseteq[{\boldsymbol{p}}]$. ∎

We consider sets of the form $\text{MT}(\boldsymbol{\pi})$ in the role of product forcing notions. A set $D\subseteq\text{MT}(\boldsymbol{\pi})$ is:

• $-$

open in $\text{MT}(\boldsymbol{\pi})$, if for any ${\boldsymbol{p}}\leqslant{\boldsymbol{q}}$ in $\text{MT}(\boldsymbol{\pi})$, ${\boldsymbol{q}}\in D\mathbin{\,\Longrightarrow\,}{\boldsymbol{p}}\in D$;

• $-$

dense in $\text{MT}(\boldsymbol{\pi})$, if for any ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, there is ${\boldsymbol{q}}\in D$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$;

• $-$

pre-dense in $\text{MT}(\boldsymbol{\pi})$, if the set $D^{\prime}=\{\hskip 0.43057pt{{\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\exists\,}{\boldsymbol{q}}\in D\,({\boldsymbol{p}}\leqslant{\boldsymbol{q}})\hskip 0.43057pt\}$ is dense in $\text{MT}(\boldsymbol{\pi})$.

Remark 4.3**.**

As a forcing notion, each $\text{MT}(\boldsymbol{\pi})$ adds an array $\langle x_{\xi k}\rangle_{\langle\xi,k\rangle\in|\boldsymbol{\pi}|}$ of reals, where each real $x_{\xi k}\in 2^{\omega}$ is a ${{\boldsymbol{\pi}}(\xi,k)}$-generic real. Namely if a set $G\subseteq\text{MT}(\boldsymbol{\pi})$ is generic over the ground set universe ${\mathbf{V}}$ then each factor

[TABLE]

(where $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$) is accordingly a set ${{\boldsymbol{\pi}}(\xi,k)}$-generic over ${\mathbf{V}}$, the real $x_{\xi k}=x_{\xi k}[G]=x[G(\xi,k)]\in 2^{\omega}$ is the only

real satisfying $x_{\xi k}\in\bigcap_{T\in G(\xi,k)}[T]$, and $x_{\xi k}$ is ${{\boldsymbol{\pi}}(\xi,k)}$-generic over ${\mathbf{V}}$ as in Lemma 2.7. ∎

The reals of the form $x_{\xi k}[G]$ will be called principal generic reals in ${\mathbf{V}}[G]$.

Definition 4.4**.**

A componentwise union of multiforcings $\boldsymbol{\pi},\text{\leavevmode{}}$ is a multiforcing $\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}}$ satisfying

$|(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})|=|\boldsymbol{\pi}|\cup|\text{\leavevmode{}}|$ and

[TABLE]

Similarly, if $\vv\boldsymbol{\pi}=\langle{\boldsymbol{\pi}}_{\alpha}\rangle_{\alpha<\lambda}$ is a sequence of multiforcings then define a multiforcing $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}\vv\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\lambda}{\boldsymbol{\pi}}_{\alpha}$ so that

$|\boldsymbol{\pi}|=\bigcup_{\alpha<\lambda}|{\boldsymbol{\pi}}_{\alpha}|$ and if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ then ${\boldsymbol{\pi}}(\xi,k)=\bigcup_{\alpha<\lambda,\>\langle\xi,k\rangle\in|{\boldsymbol{\pi}}_{\alpha}|}{{\boldsymbol{\pi}}_{\alpha}}(\xi,k)$. ∎

5 Multisystems

The next definition introduces multisystems, a multi version of the splitting/fusion technique of Section 3, whose intention is to define suitable multiforcings, as will be shown in Section 3 below.

Definition 5.1**.**

A multisystem is any map $\boldsymbol{\varphi}:|\boldsymbol{\varphi}|\to\text{\bf FSS}$, such that $|\boldsymbol{\varphi}|\subseteq\omega_{1}\times\omega\times\omega$ is finite. This amounts to

1. (1)

the map $h^{\boldsymbol{\varphi}}(\xi,k,m)=\text{\tt hgt}(\boldsymbol{\varphi}(\xi,k,m)):|\boldsymbol{\varphi}|\to\omega$, and 2. (2)

the finite collection of trees $T^{\boldsymbol{\varphi}}_{\xi k,m}(s)=\boldsymbol{\varphi}(\xi,k,m)(s)$, where $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ and $s\in 2^{\leq h^{\boldsymbol{\varphi}}(\xi,k,m)}$, such that if $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ then $\boldsymbol{\varphi}(\xi,k,m)=\langle T^{\boldsymbol{\varphi}}_{\xi k,m}(s)\rangle_{s\in 2^{\leq h^{\boldsymbol{\varphi}}(\xi,k,m)}}$ is a finite splitting system in FSS.

If $\boldsymbol{\pi}$ is a multiforcing, $|\boldsymbol{\varphi}|\subseteq{(|\boldsymbol{\pi}|)}\times\omega$, and $\boldsymbol{\varphi}(\xi,k,m)\in\text{\bf FSS}({\boldsymbol{\pi}}(\xi,k))$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ (or equivalently, $T^{\boldsymbol{\varphi}}_{\xi k,m}(s)\in{\boldsymbol{\pi}}(\xi,k)$ whenever \langle\xi,k,m\rangle\in$$\boldsymbol{\varphi} and $s\in 2^{\leq h^{\boldsymbol{\varphi}}(\xi,k,m)}$), then say that $\boldsymbol{\varphi}$ is a ${\boldsymbol{\pi}}$-multisystem, $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$. ∎

Let $\boldsymbol{\varphi},\boldsymbol{\psi}$ be multisystems. Say that $\boldsymbol{\varphi}$ extends $\boldsymbol{\psi}$, symbolically $\boldsymbol{\psi}\preccurlyeq\boldsymbol{\varphi}$, if $|\boldsymbol{\psi}|\subseteq|\boldsymbol{\varphi}|$, and, for every $\langle\xi,k,m\rangle\in|\boldsymbol{\psi}|$, $\boldsymbol{\varphi}(\xi,k,m)$ extends $\boldsymbol{\psi}(\xi,k,m)$, that is, $h^{\boldsymbol{\varphi}}(\xi,k,m)\geq h^{\boldsymbol{\psi}}(\xi,k,m)$ and $T^{\boldsymbol{\varphi}}_{\xi k,m}(s)=T^{\boldsymbol{\psi}}_{\xi k,m}(s)$ for all $s\in 2^{\leq h^{\boldsymbol{\psi}}(\xi,k,m)}$.

It will be demonstrated in Section 3 that a suitably increasing infinite sequence $\boldsymbol{\varphi}_{0}\preccurlyeq\boldsymbol{\varphi}_{1}\preccurlyeq\boldsymbol{\varphi}_{2}\preccurlyeq\dots$ of multisystems in some $\text{MS}(\boldsymbol{\pi})$ leads to a “limit” multiforcing  with $|\text{\leavevmode{}}|=\bigcup_{n}|\boldsymbol{\varphi}_{n}|$, such that each factor ${\text{\leavevmode{}}}(\xi,k)$, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, is filled in by trees $Q_{\xi k,m}$, $m<\omega$, in such a way, that the ${(\xi,k,m)}$-components of the systems $\boldsymbol{\varphi}_{n}$ are responsible for the construction of the tree $Q_{\xi k,m}$.

The next lemma introduces different ways to extend a given multisystem.

Say that a multisystem $\varphi$ is 2wise disjoint if $[T^{\varphi}_{\xi k,m}(s)]\cap[T^{\varphi}_{\eta\ell,n}(t)]=\varnothing$ for all triples $\langle\xi,k,m\rangle\neq\langle\eta,\ell,n\rangle$ in $|\varphi|$ and all $s\in 2^{h^{\boldsymbol{\varphi}}(\xi,k,m)}$ and $t\in 2^{h^{\boldsymbol{\varphi}}(\eta,\ell,n)}$.

Lemma 5.2**.**

Let $\boldsymbol{\pi}$ be a multiforcing and $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$.

1. (i)

If $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ and $h=h^{\boldsymbol{\varphi}}(\xi,k,m)$ then the extension $\boldsymbol{\psi}$ of $\boldsymbol{\varphi}$ by $h^{\boldsymbol{\psi}}(\xi,k,m)=h+1$ and $T^{\boldsymbol{\psi}}_{\xi k,m}(s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i)=T^{\boldsymbol{\varphi}}_{\xi k,m}(s){(\to i)}$ for all $s\in 2^{h}$ and $i=0,1$, belongs to $\text{MS}(\boldsymbol{\pi})$ and $\boldsymbol{\varphi}\preccurlyeq\boldsymbol{\psi}$. 2. (ii)

If $\langle\xi,k,m\rangle\notin|\boldsymbol{\varphi}|$ then the extension $\boldsymbol{\psi}$ of $\boldsymbol{\varphi}$ by $|\boldsymbol{\psi}|=|\boldsymbol{\varphi}|\cup\{\hskip 0.43057pt\langle\xi,k,m\rangle\hskip 0.43057pt\}$, $h^{\boldsymbol{\psi}}(\xi,k,m)=0$ and $T^{\boldsymbol{\psi}}_{\xi k,m}(\Lambda)=T$, where $T\in{\boldsymbol{\pi}}(\xi,k)$ and $\Lambda$ is the empty string, belongs to $\text{MS}(\boldsymbol{\pi})$ and $\boldsymbol{\varphi}\preccurlyeq\boldsymbol{\psi}$. 3. (iii)

If $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ and a set $D\subseteq{\boldsymbol{\pi}}(\xi,k)$ is open dense in ${\boldsymbol{\pi}}(\xi,k)$ then there is a multisystem $\boldsymbol{\psi}\in\text{MT}(\boldsymbol{\pi})$ such that $|\boldsymbol{\psi}|=|\boldsymbol{\varphi}|$, $\boldsymbol{\varphi}\preccurlyeq\boldsymbol{\psi}$, and $T^{\boldsymbol{\psi}}_{\xi k,m}(s)\in D$ whenever $s\in 2^{h^{\boldsymbol{\psi}}(\xi,k,m)}\;.$ 4. (iv)

*There is a 2wise disjoint

$\boldsymbol{\psi}\in\text{MT}(\boldsymbol{\pi})$ such that $|\boldsymbol{\psi}|=|\boldsymbol{\varphi}|$ and $\boldsymbol{\varphi}\preccurlyeq\boldsymbol{\psi}\;.$*

Proof**.**

To prove (iii) first use (i) to get a multisystem $\boldsymbol{\psi}\in\text{MS}(\boldsymbol{\pi})$ with $\boldsymbol{\varphi}\preccurlyeq\boldsymbol{\psi}$ and $h^{\boldsymbol{\psi}}(\xi,k,m)=h+1$, where $h=h^{\boldsymbol{\varphi}}(\xi,k,m)$. Then replace each tree $T^{\boldsymbol{\psi}}_{\xi k,m}(s)=\boldsymbol{\psi}(\xi,k,m)(s)$, $s\in 2^{h+1}$, with a suitable tree $T^{\prime}\in D$, $T^{\prime}\subseteq T^{\boldsymbol{\psi}}_{\xi k,m}(s)$.

To prove (iv) first apply (i) to get a multisystem $\boldsymbol{\psi}\in\text{MS}(\boldsymbol{\pi})$ with $\boldsymbol{\varphi}\preccurlyeq\boldsymbol{\psi}$, $|\boldsymbol{\psi}|=|\boldsymbol{\varphi}|$, and $h^{\boldsymbol{\psi}}(\xi,k,m)=h^{\boldsymbol{\varphi}}(\xi,k,m)+1$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$. Now if $\langle\xi,k,m\rangle\neq\langle\eta,\ell,n\rangle$ are triples in $|\varphi|$ and $s\in 2^{h^{\boldsymbol{\varphi}}(\xi,k,m)+1}$, $t\in 2^{h^{\boldsymbol{\varphi}}(\eta,\ell,n)+1}$, then, by Lemma 2.3(ii), there are trees $S\in{\boldsymbol{\pi}}(\xi,k)$ and $T\in{\boldsymbol{\pi}}(\eta,\ell)$ satisfying $[S]\cap[T]=\varnothing$ and $S\subseteq T^{\boldsymbol{\psi}}_{\xi k,m}(s)$, $T\subseteq T^{\boldsymbol{\psi}}_{\eta\ell,n}(t)$. Replace the trees $T^{\boldsymbol{\psi}}_{\xi k,m}(s)$, $T\subseteq T^{\boldsymbol{\psi}}_{\eta\ell,n}(t)$ with resp. $S$, $T$. Iterate this shrinking construction for all triples $\langle\xi,k,m\rangle\neq\langle\eta,\ell,n\rangle$ and strings $s,t$ as above. ∎

II Refinements

Here we consider refinements of perfect tree forcings and multiforcings, the key technical tool of definition of various forcing notions in this paper.

1 Refining perfect tree forcings

If $T\in\text{PT}$ (a perfect tree) and $D\subseteq\text{PT}$ then $X\subseteq^{\text{\tt fin}}\bigcup D$ will mean that there is a finite set $D^{\prime}\subseteq D$ such that $T\subseteq\bigcup D^{\prime}$, or equivalently $[T]\subseteq\bigcup_{S\in D^{\prime}}[S]$.

Definition 1.1**.**

Let ${{\mathbb{P}}},{{\mathbb{Q}}}\in\text{{PTF}}$ be perfect tree forcing notions. Say that ${{\mathbb{Q}}}$ is a refinement of ${{\mathbb{P}}}$

(symbolically ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}$) if

1. (1)

the set ${{\mathbb{Q}}}$ is dense in ${{\mathbb{P}}}\cup{{\mathbb{Q}}}$: if $T\in{{\mathbb{P}}}$ then ${\exists\,}Q\in{{\mathbb{Q}}}\,(Q\subseteq T)$; 2. (2)

if $Q\in{{\mathbb{Q}}}$ then $Q\subseteq^{\text{\tt fin}}\bigcup{{\mathbb{P}}}$; 3. (3)

if $Q\in{{\mathbb{Q}}}$ and $T\in{{\mathbb{P}}}$ then $[Q]\cap[T]$ is clopen in $[Q]$ and $T\not\subseteq Q$.∎

Lemma 1.2**.**

1. (i)

If ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}$ and $S\in{{\mathbb{P}}}$, $T\in{{\mathbb{Q}}}$, then $[S]\cap[T]$ is meager in $[S]$, therefore ${{\mathbb{P}}}\cap{{\mathbb{Q}}}=\varnothing$ and ${{\mathbb{Q}}}$ is open dense in ${{\mathbb{P}}}\cup{{\mathbb{Q}}}\;;$. 2. (ii)

if ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}\sqsubset{{\mathbb{R}}}$ then ${{\mathbb{P}}}\sqsubset{{\mathbb{R}}}$, thus $\sqsubset$ is a strict partial order*;*** 3. (iii)

if $\langle{{\mathbb{P}}}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\sqsubset}$-increasing sequence in PTF and $0<\mu<\lambda$ then ${{\mathbb{P}}}=\bigcup_{\alpha<\mu}{{\mathbb{P}}}_{\alpha}\sqsubset{{\mathbb{Q}}}=\bigcup_{\mu\leq\alpha<\lambda}{{\mathbb{P}}}_{\alpha}\;;$ 4. (iv)

if $\langle{{\mathbb{P}}}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\sqsubset}$-increasing sequence in PTF and each ${{\mathbb{P}}}_{\alpha}$ is special then ${{\mathbb{P}}}=\bigcup_{\alpha<\lambda}{{\mathbb{P}}}_{\alpha}$ is a regular forcing in $\text{{PTF}}\;;$ 5. (v)

in (iv), each ${{\mathbb{P}}}_{\gamma}$ is pre-dense in ${{\mathbb{P}}}=\bigcup_{\alpha<\lambda}{{\mathbb{P}}}_{\alpha}$.

Proof**.**

(i) Otherwise there is a string $u\in S$ such that $S{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{u}\subseteq[T]\cap[S]$. But $S{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{u}\in{{\mathbb{P}}}$, which contradicts to 1.1(3).

(ii), (iii) Make use of (i) to establish 1.1(3).

(iv) To check the regularity let $S\in{{\mathbb{P}}}_{\alpha}$, $T\in{{\mathbb{P}}}_{\beta}$, $\alpha\leq\beta$. If $\alpha=\beta$ then, as ${{\mathbb{P}}}_{\alpha}$ is special, the trees $S,T$ are either a. d. or ${\subseteq}$-comparable by Lemma 2.5. If $\alpha<\beta$ then $[S]\cap[T]$ is clopen in $[T]$ by 1.1(3).

(v) Let $S\in{{\mathbb{P}}}_{\alpha}$, $\alpha\neq\gamma$. If $\alpha<\gamma$ then by 1.1(1) there is a tree $T\in{{\mathbb{P}}}_{\gamma}$, $T\subseteq S$. Now let $\gamma<\alpha$. Then $S\subseteq^{\text{\tt fin}}\bigcup{{\mathbb{P}}}_{\gamma}$ by 1.1(2), in particular, there is a tree $T\in{{\mathbb{P}}}_{\gamma}$ such that $[S]\cap[T]\neq\varnothing$. However $[S]\cap[T]$ is clopen in $[S]$ by 1.1(3). Therefore $S{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{u}\subseteq T$ for a string $u\in S$. Finally $S{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{u}\in{{\mathbb{P}}}_{\alpha}$ since ${{\mathbb{P}}}_{\alpha}\in\text{{PTF}}$. ∎

Note that if ${{\mathbb{P}}},{{\mathbb{Q}}}\in\text{{PTF}}$ and ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}$ then a dense set $D\subseteq{{\mathbb{P}}}$ is not necessarily dense or even pre-dense in ${{\mathbb{P}}}\cup{{\mathbb{Q}}}$. Yet there is a special type of refinement which preserves at least pre-density. We modify the relation $\sqsubset$ as follows.

Definition 1.3**.**

Let ${{\mathbb{P}}},{{\mathbb{Q}}}\in\text{{PTF}}$ and $D\subseteq{{\mathbb{P}}}$. Say that ${{\mathbb{Q}}}$ locks $D$ over ${{\mathbb{P}}}$,

symbolically ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{Q}}}$, if ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}$ holds and

every tree $S\in{{\mathbb{Q}}}$ satisfies $S\subseteq^{\text{\tt fin}}\bigcup D$. Then simply ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}$ is equivalent to ${{\mathbb{P}}}\sqsubset_{{{\mathbb{P}}}}{{\mathbb{Q}}}$. ∎

As we’ll see now, a locked set has to be pre-dense both before and after the refinement. The additional importance of locking refinements lies in fact that, once established, it preserves under further simple refinements, that is, $\sqsubset_{D}$ is transitive in a combination with $\sqsubset$ in the sense of (ii) of the following lemma:

Lemma 1.4**.**

1. (i)

If ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{Q}}}$ then $D$ is pre-dense in ${{\mathbb{P}}}\cup{{\mathbb{Q}}}$, and if in addition ${{\mathbb{P}}}$ is regular then $D$ is pre-dense in ${{\mathbb{P}}}$ as well*;*** 2. (ii)

if ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{Q}}}\sqsubset{{\mathbb{R}}}$ (note: the second $\sqsubset$ is not $\sqsubset_{D}$!) then ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{R}}}\;;$ 3. (iii)

if $\langle{{\mathbb{P}}}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\sqsubset}$-increasing sequence in PTF, $0<\mu<\lambda$, and ${{\mathbb{P}}}=\bigcup_{\alpha<\mu}{{\mathbb{P}}}_{\alpha}\sqsubset_{D}{{\mathbb{P}}}_{\mu}$, then ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{Q}}}=\bigcup_{\mu\leq\alpha<\lambda}{{\mathbb{P}}}_{\alpha}\;.$

Proof**.**

(i) To see that $D$ is pre-dense in ${{\mathbb{P}}}\cup{{\mathbb{Q}}}$, let $T_{0}\in{{\mathbb{P}}}\cup{{\mathbb{Q}}}$. By 1.1(1), there is a tree $T\in{{\mathbb{Q}}}$, $T\subseteq T_{0}$. Then $T\subseteq^{\text{\tt fin}}\bigcup D$, in particular, there is a tree $S\in D$ with $X=[S]\cap[T]\neq\varnothing$. However $X$ is clopen in $[T]$ by 1.1(3). Therefore, by Lemma 2.2, there is a tree $T^{\prime}\in{{\mathbb{Q}}}$ with $[T^{\prime}]\subseteq X$, thus $T^{\prime}\subseteq S\in D$ and $T^{\prime}\subseteq T\subseteq T_{0}$. We conclude that $T_{0}$ is compatible with $S\in D$ in ${{\mathbb{P}}}\cup{{\mathbb{Q}}}$.

To see that $D$ is pre-dense in ${{\mathbb{P}}}$ (assuming ${{\mathbb{P}}}$ is regular), let $S_{0}\in{{\mathbb{P}}}$. It follows from the above that $S_{0}$ is compatible with some $S\in D$, hence, $S$ and $S_{0}$ are not absolutely incompatible. It remains to use Lemma 2.5(i).

To prove (ii) on the top of Lemma 1.2(ii), let $R\in{{\mathbb{R}}}$. Then $R\subseteq^{\text{\tt fin}}\bigcup{{\mathbb{Q}}}$, but each $T\in{{\mathbb{Q}}}$ satisfies $T\subseteq^{\text{\tt fin}}\bigcup D$. The same for (iii). ∎

The existence of ${\sqsubset_{D}}$-refinements will be established below.

2 Refining multiforcings

Let $\boldsymbol{\pi},\text{\leavevmode{}}$ be multiforcings. Say that is an refinement of $\boldsymbol{\pi}$,

symbolically $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$, if $|\boldsymbol{\pi}|\subseteq|\text{\leavevmode{}}|$ and ${\boldsymbol{\pi}}(\xi,k)\sqsubset{\text{\leavevmode{}}}(\xi,k)$ whenever $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$.

Corollary 2.1** (of Lemma 1.2).**

If $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\rho}$ then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\rho}$.

If $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$ then the multiforcing $\text{MT}(\text{\leavevmode{}})$ is open dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$. ∎

Our next goal is to introduce a version of Definition 1.3 suitable for multiforcings; we expect an appropriate version of Lemma 1.4 to hold.

First of all, we accomodate the definition of the relation $\subseteq^{\text{\tt fin}}$ in Section 1 for multitrees. Namely if $\boldsymbol{u}$ is a multitree and ${\boldsymbol{D}}$ a collection of multitrees, then $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{\boldsymbol{D}}$ will mean that there is a finite set ${\boldsymbol{D}}^{\prime}\subseteq{\boldsymbol{D}}$ satisfying

1. $|\boldsymbol{v}|=|\boldsymbol{u}|$ for all $\boldsymbol{v}\in{\boldsymbol{D}}^{\prime}$, and
2. $[\boldsymbol{u}]\subseteq\bigcup_{\boldsymbol{v}\in{\boldsymbol{D}}^{\prime}}[\boldsymbol{v}]$.

Definition 2.2**.**

Let $\boldsymbol{\pi},\text{\leavevmode{}}$ be multiforcings, and $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$. Say that

• locks a set ${\boldsymbol{D}}\subseteq\text{MT}(\boldsymbol{\pi})$ over $\boldsymbol{\pi}$*, symbolically $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$

if the following condition holds:

1. $(*)$

if ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, $\boldsymbol{u}\in\text{MT}(\text{\leavevmode{}})$, $|\boldsymbol{u}|\subseteq|\boldsymbol{\pi}|$, $|\boldsymbol{u}|\cap|{\boldsymbol{p}}|=\varnothing$, then there is ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ such that ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, still $|{\boldsymbol{q}}|\cap|\boldsymbol{u}|=\varnothing$, and $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$, where

[TABLE]

Note that if ${\boldsymbol{p}},\boldsymbol{u},{\boldsymbol{D}},{\boldsymbol{q}}$ are as indicated then still $\boldsymbol{u}\cup{\boldsymbol{q}}\subseteq^{\text{\tt fin}}\bigvee{\boldsymbol{D}}$ holds via the finite set ${\boldsymbol{D}}^{\prime}=\{\hskip 0.43057pt{\boldsymbol{u}^{\prime}\cup{\boldsymbol{q}}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\boldsymbol{u}^{\prime}\in{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}\hskip 0.43057pt\}\subseteq{\boldsymbol{D}}$. Anyway the definition of $\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}$ in 2.2 looks somewhat different and more complex then the definition of $\sqsubset_{D}$ in 1.3, which reflects the fact that finite-support products of forcing notions in PTF behave differently (and in more complex way) than single perfect-tree forcings. Accordingly, the next lemma, similar to Lemma 1.4, is way harder to prove.

Lemma 2.3**.**

Let $\boldsymbol{\pi},\text{\leavevmode{}},\boldsymbol{\sigma}$ be multiforcings and ${\boldsymbol{D}}\subseteq\text{MT}(\boldsymbol{\pi})$. Then$:$

1. (i)

if $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$ then ${\boldsymbol{D}}$ is dense in $\text{MT}(\boldsymbol{\pi})$ and pre-dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})\;;$ 2. (ii)

if $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$ and ${\boldsymbol{D}}\subseteq{\boldsymbol{D}}^{\prime}\subseteq\text{MT}(\boldsymbol{\pi})$ then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}^{\prime}}\text{\leavevmode{}}\;;$ 3. (iii)

if $\boldsymbol{\pi}$ is regular, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}_{i}}\text{\leavevmode{}}$ for $i=1,\dots,n$, all sets ${\boldsymbol{D}}_{i}\subseteq\text{MT}(\boldsymbol{\pi})$ are open dense in $\text{MT}(\boldsymbol{\pi})$, and ${\boldsymbol{D}}=\bigcap_{i}{\boldsymbol{D}}_{i}$, then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}\;;$ 4. (iv)

if ${{\boldsymbol{D}}}$ is open dense in $\text{MT}(\boldsymbol{\pi})$ and $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\sigma}$ then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\boldsymbol{\sigma}\;;$ 5. (v)

if $\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\mathrel{\boldsymbol{\sqsubset}}}$-increasing sequence in MF, $0<\mu<\lambda$, $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\mu}\boldsymbol{\pi}_{\alpha}$, ${{\boldsymbol{D}}}$ is open dense in $\text{MT}(\boldsymbol{\pi})$, and $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\boldsymbol{\pi}_{\mu}$, then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}=\bigcup^{\text{\tt cw}}_{\mu\leq\alpha<\lambda}\boldsymbol{\pi}_{\alpha}$.

Proof**.**

(i) To check that ${{\boldsymbol{D}}}$ is pre-dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$, let ${\boldsymbol{r}}\in\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$. Due to the product character of $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$, we can assume that $|{\boldsymbol{r}}|\subseteq|\boldsymbol{\pi}|$. Let

[TABLE]

Then ${\boldsymbol{r}}=\boldsymbol{u}\cup{\boldsymbol{p}}$, where $\boldsymbol{u}={\boldsymbol{r}}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}X\in\text{MT}(\text{\leavevmode{}})$, ${\boldsymbol{p}}={\boldsymbol{r}}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}Y\in\text{MT}(\boldsymbol{\pi})$. As locks ${{\boldsymbol{D}}}$, there is a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ such that ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, $|{\boldsymbol{q}}|\cap|\boldsymbol{u}|=\varnothing$, and $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigcup{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$. By an easy argument, there is a multitree $\boldsymbol{u}^{\prime}\in{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$ compatible with $\boldsymbol{u}$ in $\text{MT}(\text{\leavevmode{}})$; let $\boldsymbol{w}\in\text{MT}(\text{\leavevmode{}})$, $\boldsymbol{w}\leqslant\boldsymbol{u}$, $\boldsymbol{w}\leqslant\boldsymbol{u}^{\prime}$, $|\boldsymbol{w}|=|\boldsymbol{u}^{\prime}|=|\boldsymbol{u}|$. Then the multitree ${\boldsymbol{r}}^{\prime}=\boldsymbol{w}\cup{\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi}{\lor}\text{\leavevmode{}})$ satisfies ${\boldsymbol{r}}^{\prime}\leqslant{\boldsymbol{r}}$ and ${\boldsymbol{r}}^{\prime}\leqslant\boldsymbol{u}^{\prime}\cup{\boldsymbol{q}}\in{{\boldsymbol{D}}}$.

To check that ${{\boldsymbol{D}}}$ is dense in $\text{MT}(\boldsymbol{\pi})$, suppose that ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$. Let $\boldsymbol{u}=\boldsymbol{\Lambda}$ (the empty multitree) in $(*)$ ‣ 2.2 of Definition 2.2, so that $|\boldsymbol{u}|=\varnothing$ and ${{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}={{\boldsymbol{D}}}$.

(ii) is obvious. To prove (iii), let ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, $\boldsymbol{u}\in\text{MT}(\text{\leavevmode{}})$, $|\boldsymbol{u}|\subseteq|\boldsymbol{\pi}|$, $|\boldsymbol{u}|\cap|{\boldsymbol{p}}|=\varnothing$. Iterating $(*)$ ‣ 2.2 for ${\boldsymbol{D}}_{i}$, $i=1,\dots,n$, we find a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ such that ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, $|{\boldsymbol{q}}|\cap|\boldsymbol{u}|=\varnothing$, and $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{({\boldsymbol{D}}_{i})}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$ for all $i$, where

[TABLE]

Thus there are finite sets $U_{i}\subseteq{({\boldsymbol{D}}_{i})}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$ such that $[\boldsymbol{u}]\subseteq\bigcup_{\boldsymbol{v}\in U_{i}}[\boldsymbol{v}]$ for all $i$. Using the regularity assumption and Lemma 4.2, we refine multitrees in $\bigcup_{i}U_{i}$, getting a finite set $W\subseteq\text{MT}(\boldsymbol{\pi})$ such that still $|\boldsymbol{w}|=|\boldsymbol{u}|$ for all $\boldsymbol{w}\in W$, $\bigcap_{i}\bigcup_{\boldsymbol{v}\in U_{i}}[\boldsymbol{v}]=\bigcup_{\boldsymbol{w}\in W}[\boldsymbol{w}]$, and if $i=1,\dots,n$ and $\boldsymbol{w}\in W$ then $[\boldsymbol{w}]\subseteq[\boldsymbol{v}]$ for some $\boldsymbol{v}\in U_{i}$ — therefore $\boldsymbol{w}\cup{\boldsymbol{q}}\in{\boldsymbol{D}}_{i}$. We conclude that if $\boldsymbol{w}\in W$ then $\boldsymbol{w}\cup{\boldsymbol{q}}\in{\boldsymbol{D}}$, hence $\boldsymbol{w}\in{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$. Thus $W\subseteq{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$. However $[\boldsymbol{u}]\subseteq\bigcup_{\boldsymbol{w}\in W}[\boldsymbol{w}]$ by the choice of $W$. We conclude that $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$, as required.

(iv) It follows from Corollary 2.1 that $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\sigma}$, hence it remains to check that $\boldsymbol{\sigma}$ locks ${{\boldsymbol{D}}}$ over $\boldsymbol{\pi}$. Assume that $\boldsymbol{u}\in\text{MT}(\boldsymbol{\sigma})$, $|\boldsymbol{u}|\subseteq|\boldsymbol{\pi}|$, ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, $|\boldsymbol{u}|\cap|{\boldsymbol{p}}|=\varnothing$. As $\text{\leavevmode{}}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\sigma}$, there is a finite set $U\subseteq\text{MT}(\text{\leavevmode{}})$ such that $|\boldsymbol{v}|=|\boldsymbol{u}|$ for all $\boldsymbol{v}\in U$, and $[\boldsymbol{u}]\subseteq\bigcup_{\boldsymbol{v}\in U}[\boldsymbol{v}]$. As $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$, by iterated application of $(*)$ ‣ 2.2 of Definition 2.2, we get a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ such that ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, still $|{\boldsymbol{q}}|\cap|\boldsymbol{u}|=\varnothing$, and if $\boldsymbol{v}\in U$ then $\boldsymbol{v}\subseteq^{\text{\tt fin}}\bigvee{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$, where

[TABLE]

Note finally that $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee U$ by construction, hence $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$ as well.

(v) We have to check that locks ${{\boldsymbol{D}}}$ over $\boldsymbol{\pi}$. Let $\boldsymbol{u}\in\text{MT}(\text{\leavevmode{}})$, $|\boldsymbol{u}|\subseteq|\boldsymbol{\pi}|$, ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, $|\boldsymbol{u}|\cap|{\boldsymbol{p}}|=\varnothing$. As above, there is a finite set $U\subseteq\text{MT}(\boldsymbol{\pi}_{\mu})$ such that $|\boldsymbol{v}|=|\boldsymbol{u}|$ for all $\boldsymbol{v}\in U$ and $[\boldsymbol{u}]\subseteq\bigcup_{\boldsymbol{v}\in U}[\boldsymbol{v}]$. And so on as in the proof of (iv). ∎

3 Generic refinement of a multiforcing

Here we introduce a construction, due to Jensen in its original form, which implies the existence of refinements of forcings and multiforcings, of types $\sqsubset_{D}$ and $\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}$.

Definition 3.1**.**

Suppose that $\boldsymbol{\pi}$ is a small multiforcing, and ${{\mathfrak{M}}}\in\mathrm{HC}$ is any set. (Recall that $\mathrm{HC}$ = all hereditarily countable sets.) This is the input.

The set ${{\mathfrak{M}}}^{+}$ of all sets $X\in\mathrm{HC}$, ${\in}$-definable in $\mathrm{HC}$ by formulas with sets in ${{\mathfrak{M}}}$ as parameters, is still countable. Therefore there exists a ${\preccurlyeq}$-increasing sequence $\langle\boldsymbol{\varphi}(j)\rangle_{j<\omega}$ of multisystems $\boldsymbol{\varphi}(j)\in\text{MS}(\boldsymbol{\pi})$, ${{{\mathfrak{M}}}^{+}}$-generic in the sense that it intersects any set $\Delta\subseteq\text{MS}(\boldsymbol{\pi})$, $\Delta\in{{{\mathfrak{M}}}^{+}}$, dense in $\text{MS}(\boldsymbol{\pi})$. (The density means that for any $\boldsymbol{\psi}\in\text{MS}(\boldsymbol{\pi})$ there is a multisystem $\boldsymbol{\varphi}\in\Delta$ with $\boldsymbol{\psi}\preccurlyeq\boldsymbol{\varphi}$.)

Let us fix any such a ${{{\mathfrak{M}}}^{+}}$-generic sequence ${\mathbb{\Phi}}=\langle\boldsymbol{\varphi}(j)\rangle_{j<\omega}$.

Suppose that $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ and $m<\omega$. In particular, the sequence ${\mathbb{\Phi}}$ intersects every (dense by Lemma 5.2(i),(ii)) set of the form

[TABLE]

Hence a tree ${\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(s)\in{\boldsymbol{\pi}}(\xi,k)$ can be associated to any $s\in 2^{<\omega},$ such that, for all $j$, if $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}(j)|$ and $\mathop{\tt lh}(s)\leq h^{\boldsymbol{\varphi}(j)}(\xi,k,m)$ then $T^{\boldsymbol{\varphi}(j)}_{\xi k,m}(s)={\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(s)$.

Then it follows from Lemma 3.3 that each set $\textstyle{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}=\bigcap_{h}\bigcup_{s\in 2^{h}}{\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(s)$ is a tree in PT (not necessarily in ${\boldsymbol{\pi}}(\xi,k)$), as well as the trees

[TABLE]

and obviously ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(\Lambda)$. Let ${{{\mathbb{Q}}}}^{\mathbb{\Phi}}_{\xi k}=\{\hskip 0.43057pt{{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(s)\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptm<\omega\land s\in 2^{<\omega}\hskip 0.43057pt\}$.

If $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ then let $\text{\leavevmode{}}(\xi,k)={{{\mathbb{Q}}}}^{\mathbb{\Phi}}_{\xi k}=\{\hskip 0.43057pt{{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(s)\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptm<\omega\land s\in 2^{<\omega}\hskip 0.43057pt\}$.

Finally if $\text{\leavevmode{}}=\text{\leavevmode{}}[{\mathbb{\Phi}}]$ is obtained this way from an ${{{\mathfrak{M}}}^{+}}$-generic sequence ${\mathbb{\Phi}}$ of multisystems in $\text{MS}(\boldsymbol{\pi})$, then is called an ${{{\mathfrak{M}}}}$-generic refinement of $\boldsymbol{\pi}$. ∎

Proposition 3.2** (by the countability of ${{\mathfrak{M}}}^{+}$).**

If $\boldsymbol{\pi}$ is a small multiforcing and ${{\mathfrak{M}}}\in\mathrm{HC}$ then there is an ${{{\mathfrak{M}}}}$-generic refinement of $\boldsymbol{\pi}$. ∎

Theorem 3.3**.**

If $\boldsymbol{\pi}$ is a small multiforcing, a set ${{\mathfrak{M}}}\in\mathrm{HC}$ contains $\boldsymbol{\pi}$, $|\boldsymbol{\pi}|\subseteq{{\mathfrak{M}}}$, and is an ${{{\mathfrak{M}}}}$-generic refinement of $\boldsymbol{\pi}$, then$:$

1. (i)

• is a small special multiforcing, $|\text{\leavevmode{}}|=|\boldsymbol{\pi}|$, and *$\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}\;;$

2. (ii)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ and a set $D\in{{\mathfrak{M}}}$, $D\subseteq{\boldsymbol{\pi}}(\xi,k)$ is pre-dense in ${\boldsymbol{\pi}}(\xi,k)$ then ${\boldsymbol{\pi}}(\xi,k)\sqsubset_{D}{\text{\leavevmode{}}}(\xi,k)\;;$ 3. (a)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, $m<\omega$, and $s\in 2^{<\omega}$ then ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(s)={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}{(\to s)}\;;$ 4. (b)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, $m<\omega$, and $s\in 2^{<\omega}$ then ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(s)\subseteq{\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(s)\;;$ 5. (c)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, $m<\omega$, and strings $t^{\prime}\neq t$ in $2^{<\omega}$ are ${\subseteq}$-incomparable then $[{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(t^{\prime})]\cap[{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(t)]=[{\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(t^{\prime})]\cap[{\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(t)]=\varnothing\;;$ 6. (d)

if $\langle\xi,k,m\rangle\neq\langle\eta,\ell,n\rangle$ then $[{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}]\cap[{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta\ell,n}]=\varnothing\;;$ 7. (e)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, $S\in{\text{\leavevmode{}}}(\xi,k)$ and $T\in{\boldsymbol{\pi}}(\xi,k)$ then $[S]\cap[T]$ is clopen in $[S]$ and $T\not\subseteq S$, in particular, ${\boldsymbol{\pi}}(\xi,k)\cap{\text{\leavevmode{}}}(\xi,k)=\varnothing\;;$ 8. (f)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ then the set ${\text{\leavevmode{}}}(\xi,k)$ is open dense in ${\text{\leavevmode{}}}(\xi,k)\cup{\boldsymbol{\pi}}(\xi,k)$.

If in addition $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\lambda}{\boldsymbol{\pi}}_{\alpha}$, where $\lambda<\omega_{1}$, $\langle{\boldsymbol{\pi}}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\mathrel{\boldsymbol{\sqsubset}}}$-increasing sequence of small special multiforcings, and ${{\mathfrak{M}}}$ contains $\langle{\boldsymbol{\pi}}_{\alpha}\rangle_{\alpha<\lambda}$ and all $\alpha<\lambda$, then

1. (iii)

*if $\alpha<\lambda$ then ${\boldsymbol{\pi}}_{\alpha}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$. *

Proof**.**

Let $\text{\leavevmode{}}=\text{\leavevmode{}}[{\mathbb{\Phi}}]$ be obtained from an ${{{\mathfrak{M}}}^{+}}$-generic sequence ${\mathbb{\Phi}}$ of multisystems in $\text{MS}(\boldsymbol{\pi})$, as above. We argue in the notation of Definition 3.1.

If $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ and $m<\omega$ then by construction the system of trees ${\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(s)\in{\boldsymbol{\pi}}(\xi,k)$, $s\in 2^{<\omega},$ satisfies 3.1$(*)$ ‣ 3.1 on the whole domain $s\in 2^{<\omega}.$ This leads to (a), (b) (essentially corollaries of Lemma 3.3) and (c).

To prove (d) note that the set $\Delta$ of all 2wise disjoint multisystems $\boldsymbol{\varphi}$ such that $|\boldsymbol{\varphi}|$ contains both $\langle\xi,k,m\rangle$ and $\langle\eta,\ell,n\rangle$, is dense in $\text{MS}(\boldsymbol{\pi})$ by Lemma 5.2, and obviously $\Delta\in{{{\mathfrak{M}}}^{+}}.$ Therefore there is $j<\omega$ such that $\boldsymbol{\varphi}(j)\in\Delta$. Let $h=h^{\boldsymbol{\varphi}(j)}(\xi,k,m)$ and $h^{\prime}=h^{\boldsymbol{\varphi}(j)}(\eta,\ell,n)$. Then the sets

[TABLE]

are disjoint as $\boldsymbol{\varphi}(j)\in\Delta$. However $[{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}]\subseteq A$ and $[{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta\ell,n}]\subseteq B$.

(i) It follows that the sets ${\text{\leavevmode{}}}(\xi,k)={{{\mathbb{Q}}}}^{\mathbb{\Phi}}_{\xi k}$ are special PTFs (Definition 2.4), and hence is a small special multiforcing, as in (i), and $|\text{\leavevmode{}}|=|\boldsymbol{\pi}|$.

(e) To prove the clopenness claim, note that the set $\Delta$ of all multisystems $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ such that $\langle\xi,k,m\rangle\in|\varphi|$ and if $s\in 2^{h},$ where $h=h^{\boldsymbol{\varphi}}(\xi,k,m)$, then either $T^{\boldsymbol{\varphi}}_{\xi k,m}(s)\subseteq T$ or $[T^{\boldsymbol{\varphi}}_{\xi k,m}(s)]\cap[T]=\varnothing$, is dense. To prove $T\not\subseteq S$, the set $\Delta^{\prime}$ of all multisystems $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ such that $\langle\xi,k,m\rangle\in|\varphi|$ and $T\not\subseteq\bigcup_{s\in 2^{h}}T^{\boldsymbol{\varphi}}_{\xi k,m}(s)$, where $h=h^{\boldsymbol{\varphi}}(\xi,k,m)$, is dense. Note that $\Delta,\Delta^{\prime}\in{{{\mathfrak{M}}}^{+}}$ and argue as above.

(f) Density. If $T\in{\boldsymbol{\pi}}(\xi,k)$ then the set $\Delta(T)$ of all multisystems $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$, such that $T^{\boldsymbol{\varphi}}_{\xi k,m}(\Lambda)=T$ for some $m$, is dense in $\text{MS}(\boldsymbol{\pi})$ by Lemma 5.2(ii), therefore $\varphi(j)\in\Delta(T)$ for some $j$. Then ${\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(\Lambda)=T$ for some $m<\omega$. However ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi m,k}(\Lambda)\subseteq{\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(\Lambda)$. Openness. Suppose that $S\in{\text{\leavevmode{}}}(\xi,k)$, $T\in{\text{\leavevmode{}}}(\xi,k)\cup{\boldsymbol{\pi}}(\xi,k)$, $T\subseteq S$. Then $T\notin{\boldsymbol{\pi}}(\xi,k)$ by (e). Therefore $T\in{\text{\leavevmode{}}}(\xi,k)$.

(i), continuation. To establish $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$, let $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$. We have to prove that $\boldsymbol{\pi}(\xi,k)\sqsubset\text{\leavevmode{}}(\xi,k)$. This comes down to conditions (1), (2), (3) of Definition 1.1, of which (1) follows from (f) and (3) from (e), and (2) is obvious since ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(s)\subseteq{\boldsymbol{T}}^{\mathbb{\Phi}}_{\xi k,m}(s)\in\boldsymbol{\pi}(\xi,k)$ for all $m$.

(ii) As $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$ has been checked, it remains to prove ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}\subseteq^{\text{\tt fin}}\bigcup D$ for all $m$. It follows from the pre-density of $D$ that the set $D^{\prime}=\{\hskip 0.43057pt{T\in\boldsymbol{\pi}(\xi,k)\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\exists\,}S\in D(T\subseteq S)\hskip 0.43057pt\}$ is open dense in $\boldsymbol{\pi}(\xi,k)$, and still $D^{\prime}\in{{{\mathfrak{M}}}^{+}}$. Then the set $\Delta\in{{{\mathfrak{M}}}^{+}}$ of all multisystems $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ such that $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ and $T^{\boldsymbol{\varphi}}_{\xi k,m}(s)\in D$ for all $s\in h^{\boldsymbol{\varphi}}(\xi,k,m)$, is dense in $\text{MS}(\boldsymbol{\pi})$ by Lemma 5.2(iii). Thus $\boldsymbol{\varphi}(j)\in\Delta$ for some $j$, which witnesses ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}\subseteq^{\text{\tt fin}}\bigcup D$.

(iii) We have to prove that ${\boldsymbol{\pi}}_{\alpha}(\xi,k)\sqsubset\text{\leavevmode{}}(\xi,k)$ whenever $\langle\xi,k\rangle\in|{\boldsymbol{\pi}}_{\alpha}|$. And as $\boldsymbol{\pi}(\xi,k)\sqsubset\text{\leavevmode{}}(\xi,k)$ has been checked, it suffices to prove that ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}\subseteq^{\text{\tt fin}}\bigcup{\boldsymbol{\pi}}_{\alpha}(\xi,k)$ for all $m$. However $D={\boldsymbol{\pi}}_{\alpha}(\xi,k)$ is pre-dense in $\boldsymbol{\pi}(\xi,k)$ by Lemma 1.2(v), and still $D\in{{{\mathfrak{M}}}^{+}}$, hence we can refer to (ii). ∎

Corollary 3.4**.**

In the assumptions of Proposition 3.2, if $|\boldsymbol{\pi}|\subseteq Z\subseteq\omega_{1}\times\omega$ and $Z$ is at most countable then there is a small special multiforcing such that $|\text{\leavevmode{}}|=Z$ and $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$.

Proof**.**

If $|\boldsymbol{\pi}|=Z$ then let ${{\mathfrak{M}}}$ be any countable set containing $\boldsymbol{\pi}$, pick by Proposition 3.2, and apply Theorem 3.3. If $|\boldsymbol{\pi}|\subsetneqq Z$ then we trivially extend the construction by ${\text{\leavevmode{}}}(\xi,k)={{\mathbb{P}}}_{\text{\rm coh}}$ (see Example 2.1) for all $\langle\xi,k\rangle\in Z\smallsetminus|\boldsymbol{\pi}|$. ∎

Corollary 3.5**.**

Suppose that $\lambda<\omega_{1}$, and $\langle{{\mathbb{P}}}_{\alpha}\rangle_{\alpha<\lambda}$ is an ${\sqsubset}$-increasing sequence of countable special forcings in PTF. Then there is a countable special forcing ${{\mathbb{Q}}}\in\text{{PTF}}$ such that ${{\mathbb{P}}}_{\alpha}\sqsubset{{\mathbb{Q}}}$ for each $\alpha<\lambda\;.$

Proof**.**

If $\alpha<\lambda$ then let a multiforcing ${\boldsymbol{\pi}}_{\alpha}$ be defined by $|{\boldsymbol{\pi}}_{\alpha}|=\{\hskip 0.43057pt\langle 0,0\rangle\hskip 0.43057pt\}$ and by ${{\boldsymbol{\pi}}_{\alpha}}(0,0)={{\mathbb{P}}}_{\alpha}$. By Proposition 3.2 and Theorem 3.3 there is a multiforcing  satisfying $|\text{\leavevmode{}}|=\{\hskip 0.43057pt\langle 0,0\rangle\hskip 0.43057pt\}$ and ${{\boldsymbol{\pi}}_{\alpha}}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$, ${\forall\,}\alpha$. Let ${{\mathbb{Q}}}={\text{\leavevmode{}}}(0,0)$. ∎

4 Preservation of density

This Section proves a special consequence of ${{{\mathfrak{M}}}^{+}}$-genericity of multiforcing refinements, the relation $\mathrel{\boldsymbol{\sqsubset}}$ of Definition 2.2 between a multiforcing and its refinement.

Theorem 4.1**.**

In the assumptions of Theorem 3.3, if ${{\boldsymbol{D}}}\in{{\mathfrak{M}}}^{+}$, ${{\boldsymbol{D}}}\subseteq\text{MT}(\boldsymbol{\pi})$, and ${{\boldsymbol{D}}}$ is open dense in $\text{MT}(\boldsymbol{\pi})$, then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$.

Proof**.**

We suppose that $\text{\leavevmode{}}=\text{\leavevmode{}}[{\mathbb{\Phi}}]$ is obtained from an increasing ${{{\mathfrak{M}}}^{+}}$-generic sequence ${\mathbb{\Phi}}$ of multisystems in $\text{MS}(\boldsymbol{\pi})$, as in Definition 3.1, and argue in the notation of 3.1.

Suppose that ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, $\boldsymbol{u}\in\text{MT}(\text{\leavevmode{}})$, $|\boldsymbol{u}|\cap|{\boldsymbol{p}}|=\varnothing$, as in $(*)$ ‣ 2.2 of Definition 2.2; the extra condition $|\boldsymbol{u}|\subseteq|\boldsymbol{\pi}|$ holds automatically as we still have $|\text{\leavevmode{}}|=|\boldsymbol{\pi}|$. Let $X=|\boldsymbol{u}|$, $Y=|\boldsymbol{\pi}|\smallsetminus X.$ If $\langle\xi,k\rangle\in X$ then $T^{\boldsymbol{u}}_{\xi k}={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m_{\xi k}}(s_{\xi k})$, where $m_{\xi k}<\omega$ and $s_{\xi k}\in 2^{<\omega}$. By obvious reasons we can assume that $s_{\xi k}=\Lambda$, hence $T^{\boldsymbol{u}}_{\xi k}={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m_{\xi k}}$, for all $\langle\xi,k\rangle\in X$.

Consider the set $\Delta$ of all multisystems $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ such that there is a number $H>0$ and a multitree ${\boldsymbol{q}}\in\text{MS}(\boldsymbol{\pi})$ satisfying (1), (2), (3), (4) below.

1. (1)

$|{\boldsymbol{q}}|\cap X=\varnothing$ and ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$; 2. (2)

if $\langle\xi,k\rangle\in X$ then $\langle\xi,k,m_{\xi k}\rangle\in|\boldsymbol{\varphi}|$; 3. (3)

if $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ then $h^{\boldsymbol{\varphi}}(\xi,k,m)=H$.

To formulate the last requirement, we need one more definition. Suppose that $\tau=\langle t_{\xi k}\rangle_{\langle\xi,k\rangle\in X}$ is a system of strings $\tau(\xi,k)=t_{\xi k}\in 2^{H}$, symbolically $\tau\in(2^{H}){}^{X}.$ Define a multitree $\mathbf{s}(\boldsymbol{\varphi},\tau)\in\text{MT}(\boldsymbol{\pi})$ so that $|\mathbf{s}(\boldsymbol{\varphi},\tau)|=X$ and $T^{\mathbf{s}(\boldsymbol{\varphi},\tau)}_{\xi k}=T^{\boldsymbol{\varphi}}_{\xi k,m_{\xi k}}(t_{\xi k})$ for all $\langle\xi,k\rangle\in X$. Note that $|\mathbf{s}(\boldsymbol{\varphi},\tau)|=|\boldsymbol{u}|$, and hence the multitree $\mathbf{s}(\boldsymbol{\varphi},\tau)\cup{\boldsymbol{q}}$ belongs to $\text{MT}(\boldsymbol{\pi})$ as well. 333 Here, if ${\boldsymbol{p}},{\boldsymbol{q}}$ are multitrees satisfying $|{\boldsymbol{p}}|\cap|{\boldsymbol{q}}|=\varnothing$ (disjoint domains), then ${\boldsymbol{p}}\cup{\boldsymbol{q}}$, a disjoint union, is a multitree such that $|{\boldsymbol{p}}\cup{\boldsymbol{q}}|=|{\boldsymbol{p}}|\cup|{\boldsymbol{q}}|$ and $T^{{\boldsymbol{p}}\cup{\boldsymbol{q}}}_{\xi k}=T^{{\boldsymbol{p}}}_{\xi k}$ whenever $\langle\xi,k\rangle\in|{\boldsymbol{p}}|$ but $T^{{\boldsymbol{p}}\cup{\boldsymbol{q}}}_{\xi k}=T^{{\boldsymbol{q}}}_{\xi k}$ whenever $\langle\xi,k\rangle\in|{\boldsymbol{q}}|$.

Now goes the last condition.

1. (4)

If $\tau\in(2^{H}){}^{X}$ then $\mathbf{s}(\boldsymbol{\varphi},\tau)\cup{\boldsymbol{q}}\in{\boldsymbol{D}}$.

Lemma 4.2**.**

The set $\Delta$ is dense in $\text{MS}(\boldsymbol{\pi})$.

Proof** (Lemma).**

Suppose that $\boldsymbol{\psi}\in\text{MS}(\boldsymbol{\pi})$; we have to find a multisystem $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ with $\boldsymbol{\psi}\preccurlyeq\boldsymbol{\varphi}$. First of all, by Lemma 5.2(i)(ii) we can assume that

1. (a)

if $\langle\xi,k\rangle\in X$ then $\langle\xi,k,m_{\xi k}\rangle\in|\boldsymbol{\psi}|$; 2. (b)

there is a number $g>0$ such that $h^{\boldsymbol{\psi}}(\xi,k,m)=g$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\psi}|$.

Let $H=g+1$. Define $\boldsymbol{\chi}\in\text{MS}(\boldsymbol{\pi})$ so that $|\boldsymbol{\chi}|=|\boldsymbol{\psi}|$, and $h^{\boldsymbol{\chi}}(\xi,k,m)=H$, $T^{\boldsymbol{\chi}}_{\xi k,m}(s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i)=T^{\boldsymbol{\psi}}_{\xi k,m}(s){(\to i)}$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\psi}|$ and $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i\in 2^{H}$; then $\boldsymbol{\psi}\preccurlyeq\boldsymbol{\chi}$.

It follows from the open density of ${\boldsymbol{D}}$ that there is a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ satisfying (1), and a multisystem $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ satisfying (4) and such that still $|\boldsymbol{\varphi}|=|\boldsymbol{\psi}|$ and $h^{\boldsymbol{\varphi}}(\xi,k,m)=H$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\psi}|$, and in addition

1. (c)

if $\langle\xi,k\rangle\in X$ and $s\in 2^{H}$ then $T^{\boldsymbol{\varphi}}_{\xi k,m_{\xi k}}(s)\subseteq T^{\boldsymbol{\chi}}_{\xi k,m_{\xi k}}(s)$; 2. (d)

$T^{\boldsymbol{\varphi}}_{\xi k,m}(s)=T^{\boldsymbol{\chi}}_{\xi k,m}(s)$ for all applicable $\xi,k,m,s$ not covered by (c).

Namely to achieve (4) for one particular $\tau\in(2^{H}){}^{X},$ consider the multitree ${\boldsymbol{r}}=\mathbf{s}(\boldsymbol{\chi},\tau)\cup{\boldsymbol{p}}$. There is a multitree ${\boldsymbol{r}}^{\prime}\in{\boldsymbol{D}}$, ${\boldsymbol{r}}^{\prime}\leqslant{\boldsymbol{r}}$. Let a new multisystem $\boldsymbol{\chi}^{\prime}$ be obtained from $\boldsymbol{\chi}$ by the reassignment $T^{\boldsymbol{\chi}^{\prime}}_{\xi k,m_{\xi k}}(\tau(\xi,k))=T^{{\boldsymbol{r}}^{\prime}}_{\xi k}$ for all $\langle\xi,k\rangle\in X$. To get the input for the next step, let ${\boldsymbol{p}}^{\prime}={\boldsymbol{r}}^{\prime}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}Y,$ 444 Here ${\boldsymbol{r}}^{\prime}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}Y$ is the plain restriction of the function ${\boldsymbol{r}}^{\prime}:|{\boldsymbol{r}}^{\prime}|\to\text{PT}$ to the set $|{\boldsymbol{r}}^{\prime}|\cap Y$. so that ${\boldsymbol{r}}^{\prime}=\mathbf{s}(\boldsymbol{\chi}^{\prime},\tau)\cup{\boldsymbol{p}}^{\prime}\in{\boldsymbol{D}}$.

Now consider another $\tau^{\prime}\in(2^{H}){}^{X}$ and the multitree ${\boldsymbol{r}}^{\prime}=\mathbf{s}(\boldsymbol{\chi}^{\prime},\tau^{\prime})\cup{\boldsymbol{p}}^{\prime}$. There is ${\boldsymbol{r}}^{\prime\prime}\in{\boldsymbol{D}}$, ${\boldsymbol{r}}^{\prime\prime}\leqslant{\boldsymbol{r}}^{\prime}$. Define $\boldsymbol{\chi}^{\prime\prime}$ from $\boldsymbol{\chi}^{\prime}$ by the reassignment $T^{\boldsymbol{\chi}^{\prime\prime}}_{\xi k,m_{\xi k}}(\tau^{\prime}(\xi,k))=T^{{\boldsymbol{r}}^{\prime\prime}}_{\xi k}$ for all $\langle\xi,k\rangle\in X$. Let ${\boldsymbol{p}}^{\prime\prime}={\boldsymbol{r}}^{\prime\prime}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}Y,$ so that ${\boldsymbol{r}}^{\prime\prime}=\mathbf{s}(\boldsymbol{\chi}^{\prime\prime},\tau^{\prime})\cup{\boldsymbol{p}}^{\prime\prime}\in{\boldsymbol{D}}$.

And so on. The final multisystem and multitree of this construction will be $\boldsymbol{\varphi}$ and ${\boldsymbol{q}}$ satisfying (1), (2), (3), (4). Note that $\boldsymbol{\psi}\preccurlyeq\boldsymbol{\varphi}$, as we only amend the ${H}$-th level of $\boldsymbol{\chi}$ absent in $\boldsymbol{\psi}$. $\square$ (Lemma)

Note that $\Delta$ is defined in $\mathrm{HC}$ using sets ${\boldsymbol{D}}$, $\boldsymbol{\pi}$, ${\boldsymbol{p}}$, $X$, and the map $\langle\xi,k\rangle\to m_{\xi k}:X\to\omega$ as parameters. Now, ${\boldsymbol{D}}$, $\boldsymbol{\pi}$ belong to ${{\mathfrak{M}}}^{+}$ straightforwardly, $X$ belongs to ${{\mathfrak{M}}}^{+}$ since it is a finite subset of a set $|\boldsymbol{\pi}|\subseteq{{\mathfrak{M}}}$, and ${\boldsymbol{p}}$ belongs to ${{\mathfrak{M}}}^{+}$ by similar reasons. It follows that $\Delta$ belongs to ${{\mathfrak{M}}}^{+}$ as well.

Therefore, by the lemma and the choice of ${\mathbb{\Phi}}$, there is an index $j$ such that the multisystem $\boldsymbol{\varphi}(j)$ belongs to $\Delta$, which is witnessed by a number $H>0$ and a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ satisfying (1), (2), (3), (4) for $\boldsymbol{\varphi}(j)$ instead of $\boldsymbol{\varphi}$. To prove that $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{{\boldsymbol{D}}}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$, note that the multitrees $\mathbf{s}(\boldsymbol{\varphi}(j),\tau)\cup{\boldsymbol{q}}$, $\tau\in(2^{H}){}^{X}$, belong to ${\boldsymbol{D}}$ by (4), and easily $[\boldsymbol{u}]\subseteq\bigcup_{\tau\in(2^{H}){}^{X}}[\mathbf{s}(\boldsymbol{\varphi}(j),\tau)]$. ∎

III Structure of real names

Here we turn to some details of the structure of reals in models of ${\text{MT}(\boldsymbol{\pi})}$-generic type, $\boldsymbol{\pi}$ being a multiforcing. We are going focus on non-principal reals, i. e., those different from the principal generic reals $x_{\xi k}[G]$ (Remark 4.3). We’ll work towards the goal of making every non-principal real to be non-generic with respect to each of the factor forcing notions $\boldsymbol{\pi}(\xi,k)$.

1 Real names

Our next goal is to introduce a suitable notation related to names of reals in the context of forcing notions of the form $\text{MT}(\boldsymbol{\pi})$.

Definition 1.1**.**

A real name is any set $\mathbf{c}\subseteq\text{MT}\times(\omega\times\omega)$ such that the sets $K^{\mathbf{c}}_{ni}=\{\hskip 0.43057pt{{\boldsymbol{p}}\in\text{MT}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\langle{\boldsymbol{p}},n,i\rangle\in\mathbf{c}\hskip 0.43057pt\}$ satisfy the following:

1. $(*)$

if $n,k,\ell<\omega$, $k\neq\ell$, and ${\boldsymbol{p}}\in K^{\mathbf{c}}_{nk}$, ${\boldsymbol{q}}\in K^{\mathbf{c}}_{n\ell}$, then ${\boldsymbol{p}},{\boldsymbol{q}}$ are a. d..

A real name $\mathbf{c}$ is small if each set $K^{\mathbf{c}}_{ni}$ is at most countable — then the sets $\mathop{\tt dom}\mathbf{c}=\bigcup_{n,i}K^{\mathbf{c}}_{ni}\subseteq\text{MT}$ and $|\mathbf{c}|=\bigcup_{n,i}\bigcup_{{\boldsymbol{p}}\in K^{\mathbf{c}}_{ni}}|{\boldsymbol{p}}|\subseteq\omega_{1}\times\omega$, and $\mathbf{c}$ itself, are countable, too. ∎

Definition 1.2**.**

Let $\mathbf{c}$ be a real name and $G\subseteq\text{MT}$ a pairwise compatible set. Define the evaluation $\mathbf{c}[G]\in\omega^{\omega}$ so that $\mathbf{c}[G](n)=i$ iff:

• $-$

either ${\exists\,}{\boldsymbol{p}}\in G\>{\exists\,}{\boldsymbol{q}}\in K^{\mathbf{c}}_{ni}\>({\boldsymbol{p}}\leqslant{\boldsymbol{q}})$ (recall that ${\boldsymbol{p}}\leqslant{\boldsymbol{q}}$ means ${\boldsymbol{p}}$ is stronger),

• $-$

or just $i=0$ and $\neg\;{\exists\,}{\boldsymbol{p}}\in G\>{\exists\,}{\boldsymbol{q}}\in\bigcup_{i}K^{\mathbf{c}}_{ni}\>({\boldsymbol{p}}\leqslant{\boldsymbol{q}})$ (default case). ∎

Definition 1.3**.**

Let $\boldsymbol{\pi}$ be a multiforcing. A real name $\mathbf{c}$ is said to be a ${\boldsymbol{\pi}}$-real name if, in addition to $(*)$ ‣ 1.1 above, the following condition holds:

1. $(†)$

each set $K^{\mathbf{c}}_{n}=\bigcup_{i}K^{\mathbf{c}}_{ni}$ is pre-dense for $\text{MT}(\boldsymbol{\pi})$, in the sense that the set $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}}=\{\hskip 0.43057pt{{\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\exists\,}{\boldsymbol{q}}\in K^{\mathbf{c}}_{n}\,({\boldsymbol{p}}\leqslant{\boldsymbol{q}})\hskip 0.43057pt\}$ is dense (then obviously open dense) in $\text{MT}(\boldsymbol{\pi})$.∎

Generally speaking, we do not assume that $K^{\mathbf{c}}_{n}\subseteq\text{MT}(\boldsymbol{\pi})$. However if, in addition to $(*)$ ‣ 1.1, $(†)$ ‣ 1.3 above, $K^{\mathbf{c}}_{n}\subseteq\text{MT}(\boldsymbol{\pi})$ holds for all $n$, then say that $\mathbf{c}$ is a true ${\boldsymbol{\pi}}$-real name. Then each set $K^{\mathbf{c}}_{n}=\bigcup_{i}K^{\mathbf{c}}_{ni}$ is a pre-dense subset of $\text{MT}(\boldsymbol{\pi})$.

Remark 1.4**.**

Let $\boldsymbol{\pi}$ be a multiforcing, $\mathbf{c}$ be a ${\boldsymbol{\pi}}$-real name, and a set $G\subseteq\text{MT}(\boldsymbol{\pi})$ be ${\text{MT}(\boldsymbol{\pi})}$-generic over the collection of all sets $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}}$ as in $(†)$ ‣ 1.3 (All of $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}}$ are dense by the choice of $\mathbf{c}$.) Then the “or” case in Definition 1.2 never happens as we have $G\cap(K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}})\neq\varnothing$ by the choice of $G$. ∎

Remark 1.5**.**

If $\boldsymbol{\pi}$ is a regular multiforcing then the notions of being a. d. and being incompatible in $\text{MT}(\boldsymbol{\pi})$ are equivalent by Lemma 2.5(i), so that a true ${\boldsymbol{\pi}}$-real name is the same as a ${\text{MT}(\boldsymbol{\pi})}$-name for an element of $\omega^{\omega}$ in the general theory of (unramified) forcing. ∎

Example 1.6**.**

If $\xi<\omega_{1}$, $k<\omega$, then {\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k} is a real name such that if $i=0,1$ then the set C_{ni}^{\xi k}=K^{{\overset{\text{\hskip 1.50694pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k}}_{ni} consists of a lone multitree ${\boldsymbol{r}}={\boldsymbol{r}}^{\xi k}_{ni}$ with $|{\boldsymbol{r}}|=\{\hskip 0.43057pt\langle\xi,k\rangle\hskip 0.43057pt\}$ and $T^{{\boldsymbol{r}}}_{\xi k}=\{\hskip 0.43057pt{t\in 2^{<\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\mathop{\tt lh}(t)\leq n\lor t(n)=i\hskip 0.43057pt\}$, and if $i\geq 2$ then $C_{ni}^{\xi k}=\varnothing$. ∎

Remark 1.7**.**

If $\boldsymbol{\pi}\in\text{MT}$ and $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ then {\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k} is a ${\boldsymbol{\pi}}$-real name of the real $x_{\xi k}=x_{\xi k}[G]\in 2^{\omega}$, the $(\xi,k)$th term of a ${\text{MT}(\boldsymbol{\pi})}$-generic sequence $\langle x_{\xi k}[G]\rangle_{\langle\xi,k\rangle\in|\boldsymbol{\pi}|}$. That is, if $G\subseteq\text{MT}(\boldsymbol{\pi})$ is generic then the real $x_{\xi k}[G]$ defined by 4.3 coincides with the real {\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k}[G] defined by 1.2. ∎

2 Direct forcing

The following definition of the direct forcing relation is not explicitly associated with any concrete forcing notion, but in fact the direct forcing relation (in all three instances) is compatible with any forcing notion of the form $\text{MT}(\boldsymbol{\pi})$.

Let $\mathbf{c}$ be a real name. Let us say that a multitree ${\boldsymbol{p}}$:

• •

directly forces $\mathbf{c}(n)=i$, where $n,i<\omega$, iff there is a multitree ${\boldsymbol{q}}\in K^{\mathbf{c}}_{ni}$ such that ${\boldsymbol{p}}\leqslant{\boldsymbol{q}}$ (meaning: ${\boldsymbol{p}}$ is stronger);

• •

directly forces $s\subset\mathbf{c}$, where $s\in\omega^{<\omega},$ iff for all $n<\mathop{\tt lh}(s)$, ${\boldsymbol{p}}$ directly forces $\mathbf{c}(n)=i$, where $i=s(n)$;

• •

directly forces $\mathbf{c}\notin[T]$, where $T\in\text{PT}$, iff there is a string $s\in\omega^{<\omega}\smallsetminus T$ such that ${\boldsymbol{p}}$ directly forces $s\subset\mathbf{c}$.

Lemma 2.1**.**

If $\boldsymbol{\pi}$ is a multiforcing and $\mathbf{c}$ is a ${\boldsymbol{\pi}}$-real name, ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, $T\in\text{PT}$, $n<\omega$, then

1. (i)

there is a number $i<\omega$ and a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})\hskip 0.86108pt,\linebreak[0]\>{\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, which directly forces $\mathbf{c}(n)=i\;;$ 2. (ii)

there is a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})\hskip 0.86108pt,\linebreak[0]\>{\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, which directly forces $\mathbf{c}\notin[T{(\to 0)}]$ or directly forces $\mathbf{c}\notin[T{(\to 1)}]$.

Note that if $T\in{\boldsymbol{\pi}}(\xi,k)$ then the trees $T{(\to i)}$, $i=0,1$ belong to ${\boldsymbol{\pi}}(\xi,k)$.

Proof**.**

(i) Use the density of sets $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}}$ by 1.3$(†)$ ‣ 1.3 above.

(ii) Let $r=\text{\tt stem}(T)$, $n=\mathop{\tt lh}(r)$. By (i), there is a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})\hskip 0.86108pt,\linebreak[0]\>{\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, and, for any $m\leq n$, — a number $i_{m}=0,1$, such that ${\boldsymbol{q}}$ directly forces $\mathbf{c}(m)=i_{m}$, ${\forall\,}m<n$. Let $s\in 2^{n+1}$ be defined by $s(m)=i_{m}$ for every $m\leq n$. Then ${\boldsymbol{q}}$ directly forces $s\subset\mathbf{c}$. On the other hand, $s$ cannot belong to both $T{(\to 0)}$ and $T{(\to 1)}$. ∎

3 Locking real names

The next definition extends Definition 2.2 to real names.

Definition 3.1**.**

Assume that $\boldsymbol{\pi},\text{\leavevmode{}}$ are multiforcings, $\mathbf{c}$ is a real name, and $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$. Say that

• locks $\mathbf{c}$ over $\boldsymbol{\pi}$*, symbolically $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}$,

if locks, over $\boldsymbol{\pi}$, each set $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}}$ defined in 1.3$(†)$ ‣ 1.3, in the sense of Definition 2.2. ∎

Corollary 3.2** (of Theorem 4.1).**

In the assumptions of Theorem 3.3, if ${\mathbf{c}}\in{{\mathfrak{M}}}^{+}$ and $\mathbf{c}$ is a ${\boldsymbol{\pi}}$-real name then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}$.

Proof**.**

Note that each set $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}}$ belongs to ${{\mathfrak{M}}}^{+}$ (as so do $\mathbf{c}$ and $\boldsymbol{\pi}$) and is dense in $\text{MT}(\boldsymbol{\pi})$, so it remains to apply Theorem 4.1. ∎

Lemma 3.3**.**

Let $\boldsymbol{\pi},\text{\leavevmode{}},\boldsymbol{\sigma}$ be multiforcings and $\mathbf{c}$ be a real name. Then

1. (i)

if $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}$ then $\mathbf{c}$ is a ${\boldsymbol{\pi}}$-real name and a ${(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})}$-real name*;*** 2. (ii)

if $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\sigma}$ then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\boldsymbol{\sigma}\;;$ 3. (iii)

if $\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\mathrel{\boldsymbol{\sqsubset}}}$-increasing sequence in MF, $0<\mu<\lambda$, $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\mu}\boldsymbol{\pi}_{\alpha}$, and $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\boldsymbol{\pi}_{\mu}$, then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}=\bigcup^{\text{\tt cw}}_{\mu\leq\alpha<\lambda}\boldsymbol{\pi}_{\alpha}\;.$

Proof**.**

(i) By definition, we have $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{K^{\mathbf{c}}_{n}{\hskip 0.60275pt\uparrow\hskip 0.60275pt}{\boldsymbol{\pi}}}\text{\leavevmode{}}$ for each $n$, therefore $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{\boldsymbol{\pi}}$ is dense in $\text{MT}(\boldsymbol{\pi})$ (then obviously open dense) and pre-dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$ by Lemma 2.3(i). It follows that $K^{\mathbf{c}}_{n}{\hskip 0.86108pt\uparrow\hskip 0.86108pt}{(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})}$ is dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$.

To check (ii), (ii) apply Lemma 2.3(iv),(v). ∎

4 Non-principal names and avoiding refinements

Let $\boldsymbol{\pi}$ be a multiforcing. Then $\text{MT}(\boldsymbol{\pi})$ adds a collection of reals $x_{\xi k}$, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, where each principal real $x_{\xi k}=x_{\xi k}[G]$ is ${\boldsymbol{\pi}(\xi,k)}$-generic over the ground set universe. Obviously many more reals are added, and given a ${\boldsymbol{\pi}}$-real name $\mathbf{c}$, one can elaborate different requirements for a condition ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$ to force that $\mathbf{c}$ is a name of a real of the form $x_{\xi k}$ or to force the opposite. But we are mostly interested in simple conditions related to the “opposite” part. The next definition provides such a condition.

Definition 4.1**.**

Let $\boldsymbol{\pi}$ be a multiforcing, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$. A real name  $\mathbf{c}$ is non-principal over $\boldsymbol{\pi}$ at $\xi,k$ if the following set is open dense in $\text{MT}(\boldsymbol{\pi})$:

[TABLE]

We’ll show below (Theorem 6.2(i)) that the non-principality implies $\mathbf{c}$ being not a name of the real $x_{\xi k}[{\underline{G}}]$. And further, the avoidance condition in the next definition will be shown to imply $\mathbf{c}$ being a name of a non-generic real.

Definition 4.2**.**

Let $\boldsymbol{\pi},\text{\leavevmode{}}$ be multiforcings, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$. Say that

• avoids a real name $\mathbf{c}$ over $\boldsymbol{\pi}$ at $\xi,k$*, in symbol $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$,

if for each $Q\in{\text{\leavevmode{}}}(\xi,k)$, locks, over $\boldsymbol{\pi}$, the set

[TABLE]

in the sense of Definition 2.2 — formally $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})}\text{\leavevmode{}}$. ∎

Lemma 4.3**.**

Assume that $\boldsymbol{\pi},\text{\leavevmode{}},\boldsymbol{\sigma}$ are multiforcings, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, and $\mathbf{c}$ is a ${\boldsymbol{\pi}}$-real name. Then$:$

1. (i)

if $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$ and $Q\in{\text{\leavevmode{}}}(\xi,k)$ then the set ${\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})$ is open dense in $\text{MT}(\boldsymbol{\pi})$ and pre-dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})\;;$ 2. (ii)

if $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\sigma}$ then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\boldsymbol{\sigma}\;;$ 3. (iii)

if $\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\mathrel{\boldsymbol{\sqsubset}}}$-increasing sequence in MF, $0<\mu<\lambda$, $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\mu}\boldsymbol{\pi}_{\alpha}$, and $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\boldsymbol{\pi}_{\mu}$, then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}=\bigcup^{\text{\tt cw}}_{\mu\leq\alpha<\lambda}\boldsymbol{\pi}_{\alpha}$.

Proof**.**

(i) Apply Lemma 2.3(i).

(ii) Let $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ and $S\in{\boldsymbol{\sigma}}(\xi,k)$. Then, as $\text{\leavevmode{}}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\sigma}$, there is a finite set $\{\hskip 0.43057ptQ_{1},\dots,Q_{m}\hskip 0.43057pt\}\subseteq{\text{\leavevmode{}}}(\xi,k)$ such that $S\subseteq Q_{1}\cup\dots\cup Q_{m}$. We have $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}(\mathbf{c},Q_{i},\boldsymbol{\pi})}\text{\leavevmode{}}$ for all $i$ since $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$, therefore $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}(\mathbf{c},Q_{i},\boldsymbol{\pi})}\boldsymbol{\sigma}$, ${\forall\,}i$, by Lemma 2.3(iv). Note that $\bigcap_{i}{\boldsymbol{D}}(\mathbf{c},Q_{i},\boldsymbol{\pi})\subseteq{\boldsymbol{D}}(\mathbf{c},S,\boldsymbol{\pi})$ since $S\subseteq\bigcup_{i}Q_{i}$. We conclude that $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}(\mathbf{c},S,\boldsymbol{\pi})}\boldsymbol{\sigma}$ by Lemma 2.3(ii),(iii). Therefore $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\boldsymbol{\sigma}$. ∎

5 Generic refinements avoid non-principal names

The following theorem says that generic refinements as in Section 3 avoid nonprincipal names. It resembles Theorem 4.1 to some extent, yet the latter is not directly applicable here as both the multitree $Q$ and the set ${\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})$ depend on , and hence the sets ${\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})$ do not necessarily belong to ${{\mathfrak{M}}}^{+}$. However the proof will be based on rather similar arguments.

Theorem 5.1**.**

In the assumptions of Theorem 3.3, if $\langle\eta,K\rangle\in|\boldsymbol{\pi}|\subseteq{{\mathfrak{M}}}$ and $\mathbf{c}\in{{\mathfrak{M}}}$ is a ${\boldsymbol{\pi}}$-real name non-principal over $\boldsymbol{\pi}$ at $\eta,K$ then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\eta K}\text{\leavevmode{}}$.

Proof**.**

Assume that $\text{\leavevmode{}}=\text{\leavevmode{}}[{\mathbb{\Phi}}]$ is obtained from an ${{{\mathfrak{M}}}^{+}}$-generic sequence ${\mathbb{\Phi}}$ of multisystems in $\text{MS}(\boldsymbol{\pi})$, as in Definition 3.1. We stick to the notation of 3.1.

Let $Q\in{\text{\leavevmode{}}}(\eta,K)$; we have to prove that locks the set ${\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})$ over $\boldsymbol{\pi}$. By construction $Q={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}(s_{0})\subseteq{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}$ for some ${\widetilde{m}}<\omega$; it can be assumed that $Q={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}$. Following the proof of Theorem 4.1, we suppose that ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, $\boldsymbol{u}\in\text{MT}(\text{\leavevmode{}})$, $|\boldsymbol{u}|\cap|{\boldsymbol{p}}|=\varnothing$, define $X=|\boldsymbol{u}|$, $Y=|\boldsymbol{\pi}|\smallsetminus X,$ and assume that $T^{\boldsymbol{u}}_{\xi k}={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m_{\xi k}}$, where $m_{\xi k}<\omega$, for each $\langle\xi,k\rangle\in X$.

Consider the set $\Delta$ of all multisystems $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ such that there is a number $H>0$ and a multitree ${\boldsymbol{q}}\in\text{MS}(\boldsymbol{\pi})$ satisfying conditions

1. (1)

$|{\boldsymbol{q}}|\cap X=\varnothing$ and ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}\;;$ 2. (2)

if $\langle\xi,k\rangle\in X$ then $\langle\xi,k,m_{\xi k}\rangle\in|\boldsymbol{\varphi}|$; 3. (3)

if $\langle\xi,k,m\rangle\in|\boldsymbol{\varphi}|$ then $h^{\boldsymbol{\varphi}}(\xi,k,m)=H$;

(but not (4) though) as in the proof of Theorem 4.1, along with two more requirements

1. (5)

$\langle\eta,K,{\widetilde{m}}\rangle\in|\boldsymbol{\varphi}|$ — hence still $h^{\boldsymbol{\varphi}}(\eta,K,\widetilde{m})=H$ by (3); 2. (6)

if $s\in 2^{H}$ and $\tau\in(2^{H}){}^{X}$ then $\mathbf{s}(\boldsymbol{\varphi},\tau)\cup{\boldsymbol{q}}$ directly forces $\mathbf{c}\notin[T^{\boldsymbol{\varphi}}_{\eta K,{\widetilde{m}}}(s)]$.

Lemma 5.2**.**

$\Delta$* is dense in $\text{MS}(\boldsymbol{\pi})$.*

Proof**.**

Suppose that $\boldsymbol{\psi}\in\text{MS}(\boldsymbol{\pi})$; we can assume that $\boldsymbol{\psi}$ already satisfies

1. (a)

if $\langle\xi,k\rangle\in X$ then $\langle\xi,k,m_{\xi k}\rangle\in|\boldsymbol{\psi}|$; 2. (b)

there is a number $g<\omega$ such that $h^{\boldsymbol{\psi}}(\xi,k,m)=g$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\psi}|$;

as in Lemma 4.2, and in addition $\langle\eta,K,{\widetilde{m}}\rangle\in|\boldsymbol{\psi}|$.

Let $H=g+1$. Define a multisystem $\boldsymbol{\chi}\in\text{MS}(\boldsymbol{\pi})$ so that $|\boldsymbol{\chi}|=|\boldsymbol{\psi}|$, and $h^{\boldsymbol{\chi}}(\xi,k,m)=H$, $T^{\boldsymbol{\chi}}_{\xi k,m}(s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i)=T^{\boldsymbol{\psi}}_{\xi k,m}(s){(\to i)}$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\psi}|$ and $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i\in 2^{H}$; then $\boldsymbol{\psi}\preccurlyeq\boldsymbol{\chi}$. We claim that there is a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$ satisfying (1), and a multisystem $\boldsymbol{\varphi}\in\text{MS}(\boldsymbol{\pi})$ satisfying (6) and such that still $|\boldsymbol{\varphi}|=|\boldsymbol{\psi}|$ and $h^{\boldsymbol{\varphi}}(\xi,k,m)=H$ for all $\langle\xi,k,m\rangle\in|\boldsymbol{\psi}|$, and in addition

1. (c)

if $\langle\xi,k\rangle\in X$ and $s\in 2^{H}$ then $T^{\boldsymbol{\varphi}}_{\xi k,m_{\xi k}}(s)\subseteq T^{\boldsymbol{\chi}}_{\xi k,m_{\xi k}}(s)$, and we also have $T^{\boldsymbol{\varphi}}_{\eta K,\widetilde{m}}(s)\subseteq T^{\boldsymbol{\chi}}_{\eta K,\widetilde{m}}(s)$; 2. (d)

$T^{\boldsymbol{\varphi}}_{\xi k,m}(s)=T^{\boldsymbol{\chi}}_{\xi k,m}(s)$ for all applicable $\xi,k,m,s$ not covered by (c).

To achieve (6) in one step for one particular $\tau\in(2^{H}){}^{X},$ consider the multitree ${\boldsymbol{r}}=\mathbf{s}(\boldsymbol{\chi},\tau)\cup{\boldsymbol{p}}$. By Lemma 2.1 and the density assumption of the theorem, there is a multitree ${\boldsymbol{r}}^{\prime}\in\text{MT}(\boldsymbol{\varphi})$, ${\boldsymbol{r}}^{\prime}\leqslant{\boldsymbol{r}}$, which directly forces $\mathbf{c}\notin[T^{{\boldsymbol{r}}^{\prime}}_{\eta K}]$, and there are multitrees $U_{s}\in\text{MT}(\boldsymbol{\varphi})$, $s\in 2^{H}$, such that $U_{s}\subseteq T^{\boldsymbol{\chi}}_{\eta K,\widetilde{m}}(s)$ and ${\boldsymbol{r}}^{\prime}$ directly forces $\mathbf{c}\notin[U_{s}]$, ${\forall\,}s$. Let $\boldsymbol{\chi}^{\prime}$ be obtained from $\boldsymbol{\chi}$ by the following reassignment.

1. (I)

We set $T^{\boldsymbol{\chi}^{\prime}}_{\xi k,m_{\xi k}}(\tau(\xi,k))=T^{{\boldsymbol{r}}^{\prime}}_{\xi k}$ for all $\langle\xi,k\rangle\in X$. 2. (II)

If $s\in 2^{H},$ and either $\langle\eta,K\rangle\notin X$, or ${\widetilde{m}}\neq m_{\eta K}$, or $s\neq\tau(\eta,K)$ then we set $T^{\boldsymbol{\chi}^{\prime}}_{\eta K,\widetilde{m}}(s)=U_{s}$. (Note that if $\langle\eta,K\rangle\in X$ and ${\widetilde{m}}=m_{\eta K}$ then the tree $T^{\boldsymbol{\chi}^{\prime}}_{\eta K,\widetilde{m}}(\tau(\eta,K))=T^{{\boldsymbol{r}}^{\prime}}_{\eta K}$ is already defined by (I).)

Let ${\boldsymbol{p}}^{\prime}={\boldsymbol{r}}^{\prime}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}Y,$ so that ${\boldsymbol{r}}^{\prime}=\mathbf{s}(\boldsymbol{\chi}^{\prime},\tau)\cup{\boldsymbol{p}}^{\prime}$. By construction the tree ${\boldsymbol{p}}^{\prime}$ satisfies (6), for the system $\tau$ chosen, in the case $\langle\eta,K\rangle\in X$, ${\widetilde{m}}=m_{\eta K}$, $s=\tau(\eta,K)$ by (I) and in all other cases by (II).

Now consider another $\tau^{\prime}\in(2^{H}){}^{X}$ and the multitree ${\boldsymbol{r}}^{\prime}=\mathbf{s}(\boldsymbol{\chi}^{\prime},\tau^{\prime})\cup{\boldsymbol{p}}^{\prime}$. There is a multitree ${\boldsymbol{r}}^{\prime\prime}\in\text{MT}(\boldsymbol{\pi})$, ${\boldsymbol{r}}^{\prime\prime}\leqslant{\boldsymbol{r}}^{\prime}$, which which directly forces $\mathbf{c}\notin[T^{{\boldsymbol{r}}^{\prime}}_{\eta K}]$ and $\mathbf{c}\notin[U^{\prime}_{s}]$ for each $s\in 2^{H},$ where $U^{\prime}_{s}\in\text{MT}(\boldsymbol{\varphi})$ and $U^{\prime}_{s}\subseteq T^{\boldsymbol{\chi}^{\prime}}_{\eta K,\widetilde{m}}(s)$ . Let $\boldsymbol{\chi}^{\prime\prime}$ be obtained from $\boldsymbol{\chi}^{\prime}$ by the the same reassignment (for $\tau^{\prime}$ instead of $\tau$).

And so on. The final multisystem and multitree of this construction will be $\boldsymbol{\varphi}$ and ${\boldsymbol{q}}$ satisfying (1), (2), (3), (5), (6). $\square$ (Lemma)

Come back to the theorem. Note that $\Delta\in{{{\mathfrak{M}}}^{+}},$ similarly to the proof of Theorem 4.1. Therefore, by the lemma, there is an index $j$ such that the system $\boldsymbol{\varphi}(j)$ belongs to $\Delta$. Let this be witnessed by a number $H>0$ and a multitree ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$, such that conditions (1), (2), (3), (5), (6) are satisfied for $\boldsymbol{\varphi}=\boldsymbol{\varphi}(j)$.

It remains to prove that $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{{\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$. Let $V$ consist of all multitrees $\boldsymbol{v}=\mathbf{s}(\boldsymbol{\varphi}(j),\tau)$, where $\tau\in(2^{H}){}^{X}$; $[\boldsymbol{u}]\subseteq\bigcup_{\boldsymbol{v}\in V}[\boldsymbol{v}]$ by construction.

Further, if $s\in 2^{H}$ and $\boldsymbol{v}\in V$ then $\boldsymbol{v}\cup{\boldsymbol{q}}$ directly forces $\mathbf{c}\notin[T^{\boldsymbol{\varphi}(j)}_{\eta K,\widetilde{m}}(s)]$ by (6), that is, directly forces $\mathbf{c}\notin[{\boldsymbol{T}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}(s)]$ in the notation of Definition 3.1. Therefore $\boldsymbol{v}\cup{\boldsymbol{q}}$ directly forces $\mathbf{c}\notin[{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}(s)]$ since ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}(s)\subseteq{\boldsymbol{T}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}(s)$ by Lemma 3.3(b). However $Q={\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}=\bigcup_{s\in 2^{H}}{\boldsymbol{Q}}^{\mathbb{\Phi}}_{\eta K,\widetilde{m}}(s)$ by Lemma 3.3(a). It follows that $\boldsymbol{v}\cup{\boldsymbol{q}}$ directly forces $\mathbf{c}\notin[Q]$, that is, $\boldsymbol{v}\in{{\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$.

We conclude that $V$ is a (finite) subset of ${{\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$. And this accomplishes the proof of $\boldsymbol{u}\subseteq^{\text{\tt fin}}\bigvee{{\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$. ∎

6 Consequences for reals in generic extensions

We first prove a result saying that all reals in ${\text{MT}(\boldsymbol{\pi})}$-generic extensions are adequately represented by real names. Then Theorem 6.2 will show effects of the property of being a non-principal name.

Proposition 6.1**.**

Suppose that $\boldsymbol{\pi}$ is a regular multiforcing, $G\subseteq\text{MT}(\boldsymbol{\pi})$ is generic over the ground set universe ${\mathbf{V}}$, and $x\in{\mathbf{V}}[G]\cap\omega^{\omega}$. Then

1. (i)

there is a true ${\boldsymbol{\pi}}$-real name $\mathbf{c}\in{\mathbf{V}}$ such that $x=\mathbf{c}[G]\;;$ 2. (ii)

if $\text{MT}(\boldsymbol{\pi})$ is a CCC forcing in ${\mathbf{V}}$ then there is a small true ${\boldsymbol{\pi}}$-real name $\mathbf{d}\in{\mathbf{V}}$ with $x=\mathbf{d}[G]$.

Proof**.**

(i) is an instances of a general forcing theorem (see Remark 1.5 on the effect of regularity). To prove (ii), pick a real name $\mathbf{c}$ by (i), extend each set $K^{\mathbf{c}}_{n}=\bigcup_{i}K^{\mathbf{c}}_{ni}$ to an open dense set $O_{n}$ by closing strongwards, choose maximal antichains $A_{n}\subseteq O_{n}$ in those sets — which have to be countable by CCC, and then let $A_{ni}=A_{n}\cap K^{\mathbf{c}}_{ni}$ and $\mathbf{d}=\{\hskip 0.43057pt{\langle{\boldsymbol{p}},n,i\rangle\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\boldsymbol{p}}\in A_{ni}\hskip 0.43057pt\}$. ∎

Theorem 6.2**.**

Let $\boldsymbol{\pi}$ be a regular multiforcing. Then

1. (i)

if a set $G\subseteq\text{MT}(\boldsymbol{\pi})$ is generic over the ground set universe ${\mathbf{V}}$, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, and $x\in{\mathbf{V}}[G]\cap\omega^{\omega},$ then $x\neq x_{\xi k}[G]$ if and only if there is a true ${\boldsymbol{\pi}}$-real name $\mathbf{c}$, non-principal over $\boldsymbol{\pi}$ at $\xi,k$ and such that $x=\mathbf{c}[G]$. 2. (ii)

if $\mathbf{c}$ is a ${\boldsymbol{\pi}}$-real name, $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, ** is a multiforcing, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$, and a set $G\subseteq\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$ is generic over ${\mathbf{V}}$ then $\mathbf{c}[G]\notin\bigcup_{Q\in\text{\leavevmode{}}(\xi,k)}[Q]$.

Proof**.**

(i) Suppose that $x\neq x_{\xi k}[G]$. By a known forcing theorem, there is a true ${\boldsymbol{\pi}}$-real name $\mathbf{c}$ such that $x=\mathbf{c}[G]$ and $\text{MT}(\boldsymbol{\pi})$ forces that $\mathbf{c}\neq x_{\xi k}[{\underline{G}}]$. It remains to show that $\mathbf{c}$ is a non-principal name over $\boldsymbol{\pi}$ at $\xi,k$. We have to prove that the set

[TABLE]

is open dense in $\text{MT}(\boldsymbol{\pi})$. The openness is clear, let us prove the density. Consider an arbitrary ${\boldsymbol{q}}\in\text{MT}(\boldsymbol{\pi})$. Then ${\boldsymbol{q}}$ ${\text{MT}(\boldsymbol{\pi})}$-forces $\mathbf{c}\neq x_{\xi k}[{\underline{G}}]$ by the choice of $\mathbf{c}$, hence we can assume that, for some $n$, it is ${\text{MT}(\boldsymbol{\pi})}$-forced by ${\boldsymbol{q}}$ that $\mathbf{c}(n)\neq x_{\xi k}[{\underline{G}}](n)$. Then by Lemma 2.1(i) there is a multitree ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$, ${\boldsymbol{p}}\leqslant{\boldsymbol{q}}$, and a string $s\in\omega^{n+1},$ such that ${\boldsymbol{p}}$ ${\text{MT}(\boldsymbol{\pi})}$-forces $s\subseteq\mathbf{c}$. Now it suffices to show that $s\notin T^{{\boldsymbol{p}}}_{\xi k}$. Suppose otherwise: $s\in T^{{\boldsymbol{p}}}_{\xi k}$. Then the tree $T={T^{{\boldsymbol{p}}}_{\xi k}}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}$ still belongs to $\text{MT}(\boldsymbol{\pi})$. Therefore the multitree ${\boldsymbol{r}}$ defined by $T^{{\boldsymbol{r}}}_{\xi k}=T$ and $T^{{\boldsymbol{r}}}_{\xi^{\prime}k^{\prime}}=T^{{\boldsymbol{p}}}_{\xi^{\prime}k^{\prime}}$ for each pair $\langle\xi^{\prime},k^{\prime}\rangle\neq\langle\xi,k\rangle$, belongs to $\text{MT}(\boldsymbol{\pi})$ and satisfies ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}\leqslant{\boldsymbol{q}}$. However ${\boldsymbol{r}}$ directly forces both $\mathbf{c}(n)$ and $x_{\xi k}[{\underline{G}}](n)$ to be equal to one and the same value $\ell=s(n)$, which contradicts to the choice of $n$.

To prove the converse let $\mathbf{c}\in{\mathbf{V}}$ be a real name non-principal over $\boldsymbol{\pi}$ at $\xi,k$, and $x=\mathbf{c}[G]$. Assume to the contrary that $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ and $x=x_{\xi k}[G]$. There is a multitree ${\boldsymbol{q}}\in G$ which ${\text{MT}(\boldsymbol{\pi})}$-forces $\mathbf{c}=x_{\xi k}[{\underline{G}}]$. As $\mathbf{c}$ is non-principal, there is a stronger multitree ${\boldsymbol{p}}\in G\cap{\boldsymbol{D}}^{\boldsymbol{\pi}}_{\xi k}(\mathbf{c})$, ${\boldsymbol{p}}\leqslant{\boldsymbol{q}}$. Thus ${\boldsymbol{p}}$ directly forces $\mathbf{c}\notin[T^{{\boldsymbol{p}}}_{\xi k}]$, and hence ${\text{MT}(\boldsymbol{\pi})}$-forces the same statement. Yet ${\boldsymbol{p}}$ ${\text{MT}(\boldsymbol{\pi})}$-forces {\overset{\text{\hskip 2.15277pt{}_{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k}\in[T^{{\boldsymbol{p}}}_{\xi k}], of course, and this is a contradiction.

(ii) Suppose towards the contrary that $Q\in\text{\leavevmode{}}(\xi,k)$ and $\mathbf{c}[G]\in[Q]$. By definition, locks, over $\boldsymbol{\pi}$, the set

[TABLE]

Therefore in particular ${\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})$ is pre-dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}})$ by Lemma 2.3. We conclude that $G\cap{\boldsymbol{D}}(\mathbf{c},Q,\boldsymbol{\pi})\neq\varnothing$. In other words, there is a multitree ${\boldsymbol{r}}\in\text{MT}(\boldsymbol{\pi})$ which directly forces $\mathbf{c}\notin[Q]$. It easily follows that $\mathbf{c}[G]\notin[Q]$, which is a contradiction. ∎

7 Combining refinement types

Here we summarize the properties of generic refinements considered above. The next definition combines the refinement types $\sqsubset_{D}$, $\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}$, $\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}$.

Definition 7.1**.**

Suppose that $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$ are multiforcings and ${{\mathfrak{M}}}\in\mathrm{HC}$ is any set. Let $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}$

mean that the four following requirements hold:

1. (1)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, $D\in{{\mathfrak{M}}}$, $D\subseteq\boldsymbol{\pi}(\xi,k)$, $D$ is pre-dense in $\boldsymbol{\pi}(\xi,k)$, then $\boldsymbol{\pi}(\xi,k)\sqsubset_{D}\text{\leavevmode{}}(\xi,k)$; 2. (2)

if ${{\boldsymbol{D}}}\in{{\mathfrak{M}}}$, ${{\boldsymbol{D}}}\subseteq\text{MT}(\boldsymbol{\pi})$, ${{\boldsymbol{D}}}$ is open dense in $\text{MT}(\boldsymbol{\pi})$, then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$; 3. (3)

if $\mathbf{c}\in{{\mathfrak{M}}}$ is a ${\boldsymbol{\pi}}$-real name then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}$; 4. (4)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ and $\mathbf{c}\in{{\mathfrak{M}}}$ is a ${\boldsymbol{\pi}}$-real name, non-principal over $\boldsymbol{\pi}$ at $\xi,k$, then $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$.∎

Corollary 7.2** (of lemmas 1.4, 2.3, 3.3, 4.3).**

Let $\boldsymbol{\pi},\text{\leavevmode{}},\boldsymbol{\sigma}$ be multiforcings and ${{\mathfrak{M}}}$ be a countable set. Then$:$

1. (i)

if $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\sigma}$ then $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\boldsymbol{\sigma}\;;$ 2. (ii)

if $\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ is a ${\mathrel{\boldsymbol{\sqsubset}}}$-increasing sequence in MF, $0<\mu<\lambda$, $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\mu}\boldsymbol{\pi}_{\alpha}$, and $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\boldsymbol{\pi}_{\mu}$, then $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}=\bigcup^{\text{\tt cw}}_{\mu\leq\alpha<\lambda}\boldsymbol{\pi}_{\alpha}$.∎

Corollary 7.3**.**

If $\boldsymbol{\pi}$ is a small multiforcing, ${{\mathfrak{M}}}\in\mathrm{HC}$, and is an ${{{\mathfrak{M}}}}$-generic refinement of $\boldsymbol{\pi}$ (exists by Proposition 3.2!), then $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}$.

Proof**.**

We have $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}$ by a combination of 3.3(ii), 4.1, 3.2, and 5.1. ∎

IV The forcing notion

In this chapter we define the forcing notion to prove the main theorem. It will have the form $\text{MT}(\nmi\Pi)$, for a certain multiforcing $\nmi\Pi$ with $|\nmi\Pi|=\omega_{1}\times\omega$. The multiforcing $\nmi\Pi$ will be equal to the componentwise union of terms of a certain increasing sequence $\vv{\nmi\Pi}$ of small multiforcings. And quite a complicated construction of this sequence in ${\mathbf{L}}$ will make use of some ideas related to diamond-style constructions, as well as to some sort of definable genericity.

1 Increasing sequences of small multiforcings

Recall that MF is the set of all multiforcings (Section 4).

Let $\text{sMF}\subseteq\text{MF}$ be the set of all

small special multiforcings; s accounts for both small and special. Thus a multiforcing $\boldsymbol{\pi}\in\text{MF}$ belongs to sMF if $|\boldsymbol{\pi}|$ is (at most) countable and if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$ then $\boldsymbol{\pi}(\xi,k)$ is a small special (Definition 2.4) forcing in PTF.

Definition 1.1**.**

Let $\vv\text{sMF}$, resp., $\vv\text{sMF}_{\omega_{1}}$ be the set of all ${\mathrel{\boldsymbol{\sqsubset}}}$-increasing sequences $\vv\boldsymbol{\pi}=\langle{\boldsymbol{\pi}}_{\alpha}\rangle_{\alpha<\kappa}$ of

multiforcings ${\boldsymbol{\pi}}_{\alpha}\in\text{sMF}$, of length $\kappa=\mathop{\tt dom}(\vv\boldsymbol{\pi})<\omega_{1}$, resp., $\kappa=\omega_{1}$, which are domain-continuous, in the sense that if $\lambda<\kappa$ is a limit ordinal then $|{\boldsymbol{\pi}}_{\lambda}|=\bigcup_{\alpha<\lambda}|{\boldsymbol{\pi}}_{\alpha}|$. Sequences in $\vv\text{sMF}\cup\vv\text{sMF}_{\omega_{1}}$ are called multisequences. We order $\vv\text{sMF}\cup\vv\text{sMF}_{\omega_{1}}$ by the usual relations $\subseteq$ and $\subset$ of extension of sequences.

• •

Thus $\vv\boldsymbol{\pi}\subset\vv\text{\leavevmode{}}$ iff $\kappa=\mathop{\tt dom}(\vv\boldsymbol{\pi})<\lambda=\mathop{\tt dom}(\vv\text{\leavevmode{}})$ and ${\boldsymbol{\pi}}_{\alpha}={\text{\leavevmode{}}}_{\alpha}$ for all $\alpha<\kappa$.

• •

In this case, if ${{\mathfrak{M}}}$ is any set, and ${\text{\leavevmode{}}}_{\kappa}$ (the first term of $\vv\text{\leavevmode{}}$ absent in $\vv\boldsymbol{\pi}$) satisfies $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}{\text{\leavevmode{}}}_{\kappa}$, where $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\kappa}{\boldsymbol{\pi}}_{\alpha}$, then we write $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$.

If $\vv\boldsymbol{\pi}$ is a multisequence in $\vv\text{sMF}\cup\vv\text{sMF}_{\omega_{1}}$ then let $\text{MT}(\vv\boldsymbol{\pi})=\text{MT}(\boldsymbol{\pi})$, where $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}\vv\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\kappa}{\boldsymbol{\pi}}_{\alpha}$ (componentwise union), and $\kappa=\mathop{\tt dom}\vv\boldsymbol{\pi}$. Accordingly a (true) ${\vv\boldsymbol{\pi}}$-real name will mean a (true) ${\boldsymbol{\pi}}$-real name. ∎

Corollary 1.2**.**

Suppose that $\kappa<\lambda<\omega_{1}$, ${{\mathfrak{M}}}$ is a countable set, and $\vv\boldsymbol{\pi}=\langle{\boldsymbol{\pi}}_{\alpha}\rangle_{\alpha<\kappa}$ is a multisequence in $\vv\text{sMF}$. Then$:$

1. (i)

the componentwise union $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}\vv\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\kappa}{\boldsymbol{\pi}}_{\alpha}$ is a regular multiforcing*;*** 2. (ii)

there is a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ satisfying $\mathop{\tt dom}(\vv\text{\leavevmode{}})=\lambda$ and $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}\;;$ 3. (iii)

if moreover $\langle s_{\alpha}\rangle_{\alpha<\lambda}$ is an ${\subset}$-increasing sequence of countable sets $s_{\alpha}\subseteq\omega_{1}\times\omega$, $s_{\alpha}=|{\boldsymbol{\pi}}_{\alpha}|$ for all $\alpha<\kappa$, and $s_{\gamma}=\bigcup_{\alpha<\gamma}s_{\alpha}$ for all limit $\gamma<\lambda$, then there is a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ satisfying $\mathop{\tt dom}(\vv\text{\leavevmode{}})=\lambda$, $|{\text{\leavevmode{}}}_{\alpha}|=s_{\alpha}$ for all $\alpha<\lambda$, and $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}\;;$ 4. (iv)

if $\vv\boldsymbol{\pi},\vv\boldsymbol{\rho},\vv\text{\leavevmode{}}\in\vv\text{sMF}$ and $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\boldsymbol{\rho}\subseteq\vv\text{\leavevmode{}}$ then $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}\;;$ 5. (v)

if $\vv\text{\leavevmode{}}=\langle{\text{\leavevmode{}}}_{\alpha}\rangle_{\alpha<\lambda}\in\vv\text{sMF}$ and $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$ then $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\kappa}{\boldsymbol{\pi}}_{\alpha}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}{\text{\leavevmode{}}}_{\beta}$ whenever $\lambda\leq\beta<\mu$, and also $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}^{\prime}=\bigcup^{\text{\tt cw}}_{\lambda\leq\beta<\mu}{\text{\leavevmode{}}}_{\beta}$, therefore

1. (a)

$\text{MT}(\text{\leavevmode{}}^{\prime})$* is open dense in $\text{MT}(\vv\text{\leavevmode{}})\,,$* 2. (b)

if $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$, $D\in{{\mathfrak{M}}}$, $D\subseteq\boldsymbol{\pi}(\xi,k)$, $D$ is pre-dense in $\boldsymbol{\pi}(\xi,k)$, then $D$ remains pre-dense in $\boldsymbol{\pi}(\xi,k)\cup\text{\leavevmode{}}(\xi,k)\,,$ 3. (c)

*if ${{\boldsymbol{D}}}\in{{\mathfrak{M}}}$, ${{\boldsymbol{D}}}\subseteq\text{MT}(\vv\boldsymbol{\pi})$, ${{\boldsymbol{D}}}$ is open dense in $\text{MT}(\vv\boldsymbol{\pi})$, then ${{\boldsymbol{D}}}$ is pre-dense in $\text{MT}(\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}}^{\prime})=\text{MT}(\vv\text{\leavevmode{}})\,.$ *

Proof**.**

(i) Make use of Lemma 1.2(iv).

(ii) We define terms ${\text{\leavevmode{}}}_{\alpha}$ of the multisequence  required by induction.

Naturally put ${\text{\leavevmode{}}}_{\alpha}={\boldsymbol{\pi}}_{\alpha}$ for each $\alpha<\kappa$.

Now suppose that $\kappa\leq\gamma<\lambda$, multiforcings ${\text{\leavevmode{}}}_{\alpha}$, $\alpha<\gamma$, are defined, and $\vv\boldsymbol{\rho}=\langle{\text{\leavevmode{}}}_{\alpha}\rangle_{\alpha<\gamma}$ is a multisequence in $\vv\text{sMF}$. To define ${\text{\leavevmode{}}}_{\gamma}$, assume first that $\gamma$ is limit. Let $\boldsymbol{\rho}=\bigcup^{\text{\tt cw}}\vv\boldsymbol{\rho}=\bigcup^{\text{\tt cw}}_{\alpha<\gamma}{\text{\leavevmode{}}}_{\alpha}$ (componentwise union). We can assume that ${{\mathfrak{M}}}$ contains $\vv\boldsymbol{\rho}$ and satisfies $\gamma\subseteq{{\mathfrak{M}}}$ (otherwise take a bigger set). By Proposition 3.2, there is an ${{{\mathfrak{M}}}}$-generic refinement of $\boldsymbol{\rho}$. By Theorem 3.3, is small special multiforcing, $\boldsymbol{\rho}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$, and ${\boldsymbol{\rho}}_{\alpha}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$ for all $\alpha<\gamma$. In addition $\boldsymbol{\rho}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}$ by Corollary 7.3. We let ${\boldsymbol{\rho}}_{\gamma}=\text{\leavevmode{}}$. The extended multisequence $\vv\boldsymbol{\rho}_{+}=\langle{\boldsymbol{\rho}}_{\alpha}\rangle_{\alpha<\gamma+1}$ belongs to $\vv\text{sMF}$ and satisfies $\vv\boldsymbol{\rho}\subset_{{\mathfrak{M}}}\vv\boldsymbol{\rho}_{+}$.

(iii) The proof is similar, with the extra care of $|{\text{\leavevmode{}}}_{\alpha}|=s_{\alpha}$.

To prove the main claim of (v) make use of Corollary 7.2.

(iv) The relation $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$ involves only the first term of $\vv\text{\leavevmode{}}$ absent in $\vv\boldsymbol{\pi}$.

To prove (v)(a) apply Corollary 2.1.

(v)(b) As $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}^{\prime}$ and $D\in{{\mathfrak{M}}}$, we have $\boldsymbol{\pi}(\xi,k)\sqsubset_{D}\text{\leavevmode{}}(\xi,k)$. Therefore $D$ is pre-dense in $\text{\leavevmode{}}(\xi,k)$ by Lemma 1.4(ii).

(v)(c) Similarly $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{D}}$ is pre-dense in $\text{MT}(\vv\text{\leavevmode{}})$ by Lemma 2.3(i). ∎

Our plan regarding the forcing notion for Theorem 1.1 will be to define a certain multisequence $\vv{\nmi\Pi}$ in $\vv\text{sMF}_{\omega_{1}}$ and the ensuing multiforcing $\nmi\Pi=\bigcup^{\text{\tt cw}}\vv{\nmi\Pi}$ with remarkable properties related to definability and its own genericity of some sort. But we need first to introduce an important notion involved in the construction.

2 Layer restrictions of multiforcings and deciding sets

The construction of the mentioned multiforcing $\nmi\Pi$ will be maintained in such a way that different layers $\langle\nmi\Pi(k,\xi)\rangle_{\xi<\omega_{1}}$, $k<\omega$, appear rather independent of each other, albeit the principal inductive parameter will be $\xi$ rather than $k$. To reflect this feature, we introduce here a suitable notation related to layer restrictions. If $m<\omega$ then, using a special “layer restriction” symbol ${}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}$ to provide a transparent distinction from the ordinary restriction ${\hskip 0.86108pt{\restriction}\hskip 0.86108pt}$, we define sets of multitrees:

[TABLE]

and, given a multiforcing $\boldsymbol{\pi}$, define $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ similarly. Accordingly if ${\boldsymbol{p}}\in\text{MT}$ then define the layer restriction ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}\in\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ so that $|{{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}|=\{\hskip 0.43057pt{\langle\xi,k\rangle\in|{\boldsymbol{p}}|\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptk<m\hskip 0.43057pt\}$ and ${{{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}}(\xi,k)={\boldsymbol{\pi}}(\xi,k)$ whenever $\langle\xi,k\rangle\in|{{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}|$. Define ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}\in\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ similarly.

The same definitions are maintained with multiforcings:

[TABLE]

and $\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, $\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, $\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ are defined similarly.

Accordingly if $\boldsymbol{\pi}\in\text{MF}$ (in particular if $\boldsymbol{\pi}\in\text{sMF}$) and $m<\omega$ then define the layer restriction ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}\in\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ (resp., $\in\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$), so that $|{\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}|=\{\hskip 0.43057pt{\langle\xi,k\rangle\in|\boldsymbol{\pi}|\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptk<m\hskip 0.43057pt\}$ and ${{\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}}(\xi,k)={\boldsymbol{\pi}}(\xi,k)$ whenever $\langle\xi,k\rangle\in|{\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}|$. Define ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}\in\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ similarly.

A similar notation applies to multisequences (Definition 1.1). If $m<\omega$ then we let $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ be the set of all multisequences in $\vv\text{sMF}$ whose all terms belong to resp. $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$. Define similarly $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ (multisequences of length $\omega_{1}$).

And further, if $\vv\boldsymbol{\pi}=\langle{\boldsymbol{\pi}}_{\alpha}\rangle_{\alpha<\kappa}\in\vv\text{sMF}$ and $m<\omega$ then define ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}=\langle{{\boldsymbol{\pi}}_{\alpha}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}\rangle_{\alpha<\kappa}\in\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, and define ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}\in\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ similarly. The same for $\vv\boldsymbol{\pi}=\langle{\boldsymbol{\pi}}_{\alpha}\rangle_{\alpha<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$

Definition 2.1**.**

Assume that $m<\omega$. A multisequence $\vv\boldsymbol{\pi}\in\vv\text{sMF}$ ${m}$-decides a set $W$ if either ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ belongs to $W$ (positive decision) or there is no multisequence $\vv\text{\leavevmode{}}\in W\cap\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ extending ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ (negative decision). ∎

Lemma 2.2**.**

If $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, ${{\mathfrak{M}}}$ is countable, $W$ is any set, and $m<\omega,$ then there is a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ such that $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$ and $\vv\text{\leavevmode{}}$ ${m}$-decides $W.$

Proof**.**

By Corollary 1.2, there is a multisequence $\vv\boldsymbol{\rho}\in\vv\text{sMF}$ such that $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\boldsymbol{\rho}$. Then either $\vv\boldsymbol{\rho}$ outright ${m}$-decides $W$ negatively, or there is a sequence $\vv\boldsymbol{\sigma}\in W\cap\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ satisfying ${\vv\boldsymbol{\rho}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\subseteq\vv\boldsymbol{\sigma}$.

On the other hand, using Corollary 1.2(iii), we get a multisequence $\vv\boldsymbol{\sigma}^{\prime}\in\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ of the same length as $\vv\boldsymbol{\sigma}$, such that ${\vv\boldsymbol{\rho}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}\subseteq\vv\boldsymbol{\sigma}^{\prime}$. Therefore there exists a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ of that same length, satisfying ${\vv\text{\leavevmode{}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}=\vv\boldsymbol{\sigma}$ and ${\vv\text{\leavevmode{}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}=\vv\boldsymbol{\sigma}^{\prime}$ — then obviously $\vv\boldsymbol{\rho}\subseteq\vv\text{\leavevmode{}}$ and by definition $\vv\text{\leavevmode{}}$ decides $W$ positively. Finally we have $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\boldsymbol{\rho}\subseteq\vv\text{\leavevmode{}}$, and hence $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$ by Corollary 1.2(iv). ∎

3 Auxiliary diamond sequences

Recall that $\mathrm{HC}$ is the set of all hereditarily countable sets (those with finite or countable transitive closures).

The next theorem employs the technique of diamond sequences in ${\mathbf{L}}$.

Theorem 3.1** (in ${\mathbf{L}}$).**

There exist $\varDelta^{\mathrm{HC}}_{1}$ sequences $\langle{\vv\boldsymbol{\pi}}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$, $\langle{D}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$, $\langle{z}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$, such that, for every $\mu$, ${D}\lceil{\mu}\rceil$ and ${z}\lceil{\mu}\rceil$ are sets in $\mathrm{HC}$, ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil\in\vv\text{sMF}$, $\mathop{\tt dom}({\vv\boldsymbol{\pi}}\lceil{\mu}\rceil)=\mu$, and in addition if $\vv{\nmi\Pi}=\langle{\nmi\Pi}_{\nu}\rangle_{\nu<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$, $z\in\mathrm{HC}$, and $D\subseteq\text{MT}(\vv{\nmi\Pi})$, then the set $M$ of all ordinals $\mu<\omega_{1}$ such that

1. (a)

${z}\lceil{\mu}\rceil=z\;;$** 2. (b)

${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil$* is equal to the restricted sub-multisequence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu=\langle{\nmi\Pi}_{\nu}\rangle_{\nu<\mu}\;;$* 3. (c)

${D}\lceil{\mu}\rceil=D\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu)\;;$**

is stationary in $\omega_{1}$.

Proof**.**

Arguing in ${\mathbf{L}}$, the constructible universe, we let $\leqslant_{{\mathbf{L}}}$ be the canonical wellordering of ${\mathbf{L}}$. It is known that $\leqslant_{{\mathbf{L}}}$ orders $\mathrm{HC}$ similarly to $\omega_{1}$, and that $\leqslant_{{\mathbf{L}}}$ is $\varDelta^{\mathrm{HC}}_{1}$ and has the goodness property: the set of all ${\leqslant_{{\mathbf{L}}}}$-initial segments $I_{x}(\leqslant_{{\mathbf{L}}})=\{\hskip 0.43057pt{y\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pty\leqslant_{{\mathbf{L}}}x\hskip 0.43057pt\}$, $x\in\mathrm{HC}$, is still $\varDelta^{\mathrm{HC}}_{1}$.

We begin with a $\varDelta^{\mathrm{HC}}_{1}$ sequence of sets $S_{\alpha}\subseteq\alpha$, $\alpha<\omega_{1}$, such that

1. (A)

if $X\subseteq\mathrm{HC}$ then the set $\{\hskip 0.43057pt{\alpha<\omega_{1}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptS_{\alpha}=X\cap\alpha\hskip 0.43057pt\}$ is stationary in $\omega_{1}$.

This is a well-known instance of the diamond principle true in ${\mathbf{L}}$. The additional definability property can be achieved by taking the ${\leqslant_{{\mathbf{L}}}}$-least possible $S_{\alpha}$ at each step $\alpha$. We get the following two results as easy corollaries.

First, let $A_{\mu}=\{\hskip 0.43057pt{c_{\alpha}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\alpha\in S_{\mu}\hskip 0.43057pt\}$, where $c_{\alpha}$ is the ${\alpha}$-th element of $\mathrm{HC}$ in the sense of the ordering $\leqslant_{{\mathbf{L}}}$. Then $\langle A_{\mu}\rangle_{\mu<\omega_{1}}$ is still a $\varDelta^{\mathrm{HC}}_{1}$ sequence, and

1. (B)

if $X_{\alpha}\in\mathrm{HC}$ for all $\alpha<\omega_{1}$ then the set $\{\hskip 0.43057pt{\mu\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptA_{\mu}=\{\hskip 0.43057pt{X_{\alpha}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\alpha<\mu\hskip 0.43057pt\}\hskip 0.43057pt\}$ is stationary in $\omega_{1}$.

Second, for any $\alpha$, if $A_{\alpha}=\langle a_{\gamma}\rangle_{\gamma<\alpha}$, where each $a_{\gamma}$ itself is equal to an ${\omega}$-sequence $\langle a^{n}_{\gamma}\rangle_{n<\omega}$, then let $B^{n}_{\alpha}=\langle a^{n}_{\gamma}\rangle_{\gamma<\alpha}$ for all $n$. Otherwise let $B^{n}_{\alpha}=\varnothing$, ${\forall\,}n$. Then \big{\langle}B_{\alpha}\big{\rangle}{}^{n<\omega}_{\alpha<\omega_{1}} is still a $\varDelta^{\mathrm{HC}}_{1}$ system of sets in $\mathrm{HC}$, such that

1. (C)

if $X^{n}_{\alpha}\in\mathrm{HC}$ for all $\alpha<\omega_{1}$, $n<\omega$, then, for every $\mu<\omega_{1}$, the set $\{\hskip 0.43057pt{\mu\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\forall\,}n\,(B^{n}_{\mu}=\{\hskip 0.43057pt{X^{n}_{\alpha}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\alpha<\mu\hskip 0.43057pt\}\hskip 0.43057pt\}$ is stationary in $\omega_{1}$.

Now things become more routinely complex.

Let $\mu<\omega_{1}$. We define ${z}\lceil{\mu}\rceil=\bigcup B^{0}_{\mu}$. If $B^{1}_{\mu}\in\vv\text{sMF}$ and $\mathop{\tt dom}(B^{1}_{\mu})=\mu$ then let ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil=B^{1}_{\mu}$; otherwise let ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil$ be equal to the ${\leqslant_{{\mathbf{L}}}}$-least multisequence in $\vv\text{sMF}$ of length $\mu$. (Those exist by Corollary 1.2(ii).) Finally we let ${D}\lceil{\mu}\rceil=\bigcup B^{2}_{\mu+1}$.

Let’s show that the sequences of sets ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil$, ${D}\lceil{\mu}\rceil$, ${z}\lceil{\mu}\rceil$ prove the theorem. Suppose that $\vv{\nmi\Pi}=\langle{\nmi\Pi}_{\nu}\rangle_{\nu<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$, $z\in\mathrm{HC}$, and $D\subseteq\text{MT}(\vv{\nmi\Pi})$. Let $X^{0}_{\alpha}=z$, $X^{1}_{\alpha}=\langle\alpha,{\nmi\Pi}_{\alpha}\rangle$, $X^{2}_{\alpha}=D\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\alpha)$ for all $\alpha$. The set

[TABLE]

is stationary by (C). Assume that $\mu\in M$. Then $B^{0}_{\mu}=\{\hskip 0.43057pt{X^{0}_{\alpha}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\alpha<\mu\hskip 0.43057pt\}=\{\hskip 0.43057ptz\hskip 0.43057pt\}$, therefore ${z}\lceil{\mu}\rceil=z$. Further $B^{1}_{\mu}=\{\hskip 0.43057pt{X^{1}_{\alpha}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\alpha<\mu\hskip 0.43057pt\}=\{\hskip 0.43057pt{\langle\alpha,{\nmi\Pi}_{\alpha}\rangle\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\alpha<\mu\hskip 0.43057pt\}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu\in\vv\text{sMF}$, therefore ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu$. Finally we have ${D}\lceil{\mu}\rceil=\bigcup B^{2}_{\mu+1}=\bigcup_{\alpha\leq\mu}{X^{2}_{\alpha}}=D\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu)$, as required. ∎

4 Key sequence theorem

Now we prove a theorem which introduces the key multisequence $\vv{\nmi\Pi}$.

Theorem 4.1** (${\mathbf{V}}={\mathbf{L}}$).**

There exists a multisequence $\vv{\nmi\Pi}=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$ satisfying the following requirements$:$

1. (i)

if $m<\omega$ then the multisequence ${\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ belongs to the class $\varDelta^{\mathrm{HC}}_{m+2}\;;$ 2. (ii)

if $m^{\prime}<\omega$ and $W\subseteq\vv\text{sMF}$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m^{\prime}+1}$ set then there is an ordinal $\gamma<\omega_{1}$ such that the multisequence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ ${m^{\prime}}$-decides $W\;;$ 3. (iii)

if a set $D\subseteq\text{MT}(\vv{\nmi\Pi})$ is dense in $\text{MT}(\vv{\nmi\Pi})$, then the set $Z$ of all ordinals $\gamma<\omega_{1}$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset_{\{\hskip 0.3014ptD\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma)\hskip 0.3014pt\}}\vv{\nmi\Pi}$, is stationary in $\omega_{1}$.

Proof**.**

If $m<\omega$ then let $\text{un}_{m}(p,x)$ be a canonical universal $\varSigma_{m+1}$ formula, so that the family of all ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m+1}$ sets $X\subseteq\mathrm{HC}$ (those definable in $\mathrm{HC}$ by $\varSigma_{m+1}$ formulas with parameters in $\mathrm{HC}$) is equal to the family of all sets of the form $\Upsilon_{m}(p)=\{\hskip 0.43057pt{x\in\mathrm{HC}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\mathrm{HC}\models\text{un}_{m}(p,x)\hskip 0.43057pt\}$, $p\in\mathrm{HC}$.

1. (I)

Fix $\varDelta^{\mathrm{HC}}_{1}$ sequences $\langle{\vv\boldsymbol{\pi}}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$, $\langle{D}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$, and $\langle{z}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$ satisfying Theorem 3.1; the terms ${D}\lceil{\mu}\rceil\hskip 0.86108pt,\linebreak[0]\hskip 0.86108pt{z}\lceil{\mu}\rceil\hskip 0.86108pt,\linebreak[0]\hskip 0.86108pt{\vv\boldsymbol{\pi}}\lceil{\mu}\rceil$ of the sequences belong to $\mathrm{HC}$, and in addition ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil\in\vv\text{sMF}$, $\mathop{\tt dom}({\vv\boldsymbol{\pi}}\lceil{\mu}\rceil)=\mu$. 2. (II)

Let $\mu<\omega_{1}$. If ${z}\lceil{\mu}\rceil$ is a pair of the form ${z}\lceil{\mu}\rceil=\langle m,p\rangle$ then let ${m}\lceil{\mu}\rceil=m$ and ${p}\lceil{\mu}\rceil=p$, otherwise let ${m}\lceil{\mu}\rceil={p}\lceil{\mu}\rceil=0$. 3. (III)

If $m<\omega$ then let, by Lemma 2.2, ${\vv\boldsymbol{\pi}}\lceil{\mu,m}\rceil\in\vv\text{sMF}$ be the ${\leqslant_{{\mathbf{L}}}}$-least multisequence in $\vv\text{sMF}$ which satisfies ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil\subset_{\{\hskip 0.3014pt{D}\lceil{\mu}\rceil\hskip 0.3014pt\}}{\vv\boldsymbol{\pi}}\lceil{\mu,m}\rceil$ and ${m}$-decides the set $\Upsilon_{m}({p}\lceil{\mu}\rceil)$. Let $\mathord{\lceil\mu,m\rceil}^{+}=\mathop{\tt dom}({\vv\boldsymbol{\pi}}\lceil{\mu,m}\rceil)$; then $\mu<\mathord{\lceil\mu,m\rceil}^{+}<\omega_{1}$.

Proposition 4.2** (in ${\mathbf{L}}$).**

The sequences $\langle{m}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$ and $\langle{p}\lceil{\mu}\rceil\rangle_{\mu<\omega_{1}}$ belong to the definability class $\varDelta^{\mathrm{HC}}_{1}$. If $m<\omega$ then the sequences $\langle{\vv\boldsymbol{\pi}}\lceil{\mu,m}\rceil\rangle_{\mu<\omega_{1}}$ and $\langle\mathord{\lceil\mu,m\rceil}^{+}\rangle_{\mu<\omega_{1}}$ belong to the class $\varDelta^{\mathrm{HC}}_{m+2}$.

Proof**.**

Routine. Note that ${\vv\boldsymbol{\pi}}\lceil{\mu,m}\rceil$ and $\mathord{\lceil\mu,m\rceil}^{+}$ depend on $m$ through the formulas $\text{un}_{m}(\cdot,\cdot)$, whose complexity strictly increases with $m\to\infty$. ∎

Now define a multisequence $\vv{\nmi\Pi}=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$ and a family of strictly increasing, continuous maps $\mu_{m}:\omega_{1}\to\omega_{1}$, $m<\omega$, as follows$:$

1. $1^{\circ}$.

Let $\mu_{m}(0)=0$ and $\mu_{m}(\lambda)=\mathop{\tt sup}_{\gamma<\lambda}\mu_{m}(\gamma)$ for all $m$ and all limit $\lambda<\omega_{1}$. 2. $2^{\circ}$.

Suppose that $m<\omega$, $\gamma<\omega_{1}$, $\mu=\mu_{m}(\gamma)$, and the twofold-restricted sequence ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}=({\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}){\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu$ is already defined. If the following holds:

1. $(*)$

$m\geq m^{\prime}={m}\lceil{\mu}\rceil$ and ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ coincides with ${{\vv\boldsymbol{\pi}}\lceil{\mu}\rceil}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$,

then let $\mu_{m}(\gamma+1)=\mathord{\lceil\mu,m^{\prime}\rceil}^{+}$ and ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{\mathord{\lceil\mu,m^{\prime}\rceil}^{+}})}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}={{\vv\boldsymbol{\pi}}\lceil{\mu,m^{\prime}}\rceil}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$. 3. $3^{\circ}$.

In the assumptions of $2^{\circ}$, if 2^{\circ}$$(*) fails, then let $\vv\boldsymbol{\rho}$ is the ${\leqslant_{{\mathbf{L}}}}$-least multisequence in $\vv\text{sMF}$ with ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}\subset\vv\boldsymbol{\rho}$ (we refer to Corollary 1.2), and define $\mu_{m}(\gamma+1)=\mathop{\tt dom}(\vv\boldsymbol{\rho})$ and ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{\mu_{m}(\gamma+1)})}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}={\vv\boldsymbol{\rho}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}\;.$

To conclude, given $\gamma<\omega_{1}$ and $m$, if an ordinal $\mu=\mu_{m}(\gamma)$, and a multisequence ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}=({\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}){\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu$ are defined, then items $2^{\circ}$, $3^{\circ}$ define a bigger ordinal $\mu_{m}(\gamma+1)>\mu=\mu_{m}(\gamma)$ and a longer multisequence ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{\mu_{m}(\gamma+1)})}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ satisfying ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}\subset{(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{\mu_{m}(\gamma+1)})}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$. Thus overall items $1^{\circ}$, $2^{\circ}$, $3^{\circ}$ of the definition contain straightforward instructions as how to uniquely define the layers ${\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ and maps $\mu_{m}$ for different $m<\omega$, independently from each other.

From now on, fix a multisequence $\vv{\nmi\Pi}=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$ of multiforcings $\nmi\Pi_{\alpha}\in\text{sMF}$ and increasing continuous maps $\mu_{m}:\omega_{1}\to\omega_{1}$ defined by $1^{\circ}$, $2^{\circ}$, $3^{\circ}$. As the maps $\mu_{m}$ are continuous, the following holds:

Proposition 4.3** (in ${\mathbf{L}}$).**

${{\mathbb{C}}}=\{\hskip 0.43057pt{\gamma<\omega_{1}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\forall\,}m\,(\gamma=\mu_{m}(\gamma))\hskip 0.43057pt\}$* is a club in $\omega_{1}$.∎*

To show that $\vv{\nmi\Pi}$ proves Theorem 4.1, we check items (i), (ii), (iii).

(i) Let $m<\omega$. Then the multisequence ${\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ and the map $\mu_{m}$ belong to the class $\varDelta^{\mathrm{HC}}_{m+2}$ by Proposition 4.2; a routine proof is omitted.

(ii) Suppose that $m^{\prime}<\omega$ and $W\subseteq\vv\text{sMF}$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m^{\prime}+2}$ set. Pick $p\in\mathrm{HC}$ such that $W=\Upsilon_{m^{\prime}}(p)$. Let $z=\langle m^{\prime},p\rangle$. As ${{\mathbb{C}}}$ is a club, it follows from the choice of terms ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil$, ${D}\lceil{\mu}\rceil$, and ${z}\lceil{\mu}\rceil$, by (I) and Theorem 3.1, that there is an ordinal $\gamma\in{{\mathbb{C}}}$ such that ${\vv\boldsymbol{\pi}}\lceil{\gamma}\rceil=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ and ${z}\lceil{\gamma}\rceil=z$ — hence, ${m}\lceil{\gamma}\rceil=m^{\prime}$ and ${p}\lceil{\gamma}\rceil=p$.

Let $\mu=\gamma$; then also $\mu=\mu_{m}(\gamma)$, ${\forall\,}m$ — since $\gamma\in{{\mathbb{C}}}$, and $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu={\vv\boldsymbol{\pi}}\lceil{\mu}\rceil$.

Then it follows from the choice of $\vv{\nmi\Pi}$ that item $2^{\circ}$ of the construction applies for the ordinal $\gamma$ chosen and all $m\geq m^{\prime}$. It follows that the multisequence $\vv\boldsymbol{\rho}={\vv\boldsymbol{\pi}}\lceil{\mu,m^{\prime}}\rceil$ and the ordinal $\nu=\mu_{m}(\gamma+1)=\mathord{\lceil\mu,m^{\prime}\rceil}^{+}$ satisfy $\nu=\mathop{\tt dom}(\vv\boldsymbol{\rho})$ and ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{\nu})}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}={\vv\boldsymbol{\rho}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ for all $m\geq m^{\prime}$. In other words, ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{\nu})}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm^{\prime}}={\vv\boldsymbol{\rho}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm^{\prime}}$.

However by definition $\vv\boldsymbol{\rho}$ ${m^{\prime}}$-decides the set $W=\Upsilon_{m^{\prime}}(p)$, and the definition of this property depends only on ${\vv\boldsymbol{\rho}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm^{\prime}}$.

(iii) Assume that a set $D\subseteq\text{MT}(\vv{\nmi\Pi})$ is dense in $\text{MT}(\vv{\nmi\Pi})$, and $C\subseteq{{\mathbb{C}}}$ is a club in $\omega_{1}$. Following the proof of (ii), we find an ordinal $\gamma\in C$ such that ${\vv\boldsymbol{\pi}}\lceil{\gamma}\rceil=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$, ${m}\lceil{\gamma}\rceil=0$, and ${D}\lceil{\gamma}\rceil=D\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)$, Note that $\gamma=\mu_{m}(\gamma)$, ${\forall\,}m$. We have ${\vv\boldsymbol{\pi}}\lceil{\gamma}\rceil\subset_{\{\hskip 0.3014pt{D}\lceil{\gamma}\rceil\hskip 0.3014pt\}}{\vv\boldsymbol{\pi}}\lceil{\gamma,0}\rceil$ by (III) (with $\mu=\gamma$), that is,

[TABLE]

Yet it follows from the choice of $\gamma$ that condition 2^{\circ}$$(*) holds (for $\mu=\gamma$) for all $m\geq 0$. Then, by definition $2^{\circ}$, the ordinal $\mu^{+}=\mathord{\lceil\gamma,m\rceil}^{+}$ satisfies $\mu^{+}=\mu_{m}(\gamma+1)$ and $(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu^{+}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}=({\vv\boldsymbol{\pi}}\lceil{\gamma,0}\rceil){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ for all $m$, that is, just $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu^{+}={\vv\boldsymbol{\pi}}\lceil{\gamma,0}\rceil$. We conclude that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset_{\{\hskip 0.3014ptD\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma)\hskip 0.3014pt\}}\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\mu^{+}$ by $({\dagger})$, therefore we have $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset_{\{\hskip 0.3014ptD\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma)\hskip 0.3014pt\}}\vv{\nmi\Pi}$, as required. $\square$ (Theorem 4.1)

Definition 4.4** (in ${\mathbf{L}}$).**

From now on we fix a multisequence $\vv{\nmi\Pi}=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$ satisfying requirements of Theorem 4.1, that is,

1. (i)

if $m<\omega$ then the multisequence ${\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ belongs to the class $\varDelta^{\mathrm{HC}}_{m+2}\;;$ 2. (ii)

if $m^{\prime}<\omega$ and $W\subseteq\vv\text{sMF}$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m^{\prime}+1}$ set then there is an ordinal $\gamma<\omega_{1}$ such that the multisequence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ ${m^{\prime}}$-decides $W\;;$ 3. (iii)

if a set $D\subseteq\text{MT}(\vv{\nmi\Pi})$ is dense in $\text{MT}(\vv{\nmi\Pi})$, then the set $Z$ of all ordinals $\gamma<\omega_{1}$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset_{\{\hskip 0.3014ptD\cap\text{MT}(\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma)\hskip 0.3014pt\}}\vv{\nmi\Pi}$, is stationary in $\omega_{1}$.

We call $\vv{\nmi\Pi}$ the key multisequence. ∎

A set $U\subseteq\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ is dense in $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ if for each $\vv\boldsymbol{\pi}\in\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ there is a multisequence $\vv\text{\leavevmode{}}\in U$ satisfying $\vv\boldsymbol{\pi}\subseteq\vv\text{\leavevmode{}}$.

Lemma 4.5**.**

If $m<\omega$ and $W\subseteq\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m+1}$ set dense in $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ then there is an ordinal $\gamma<\omega_{1}$ such that ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in W$. In particular, if $W\subseteq\vv\text{sMF}$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{1}$ set dense in $\vv\text{sMF}$ then there is $\gamma<\omega_{1}$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\in W$.

Proof**.**

Apply 4.4(ii). The negative decision is impossible by the density. ∎

5 Key product forcing

We continue to argue in ${\mathbf{L}}$, and we’ll make use of the key multisequence $\vv{\nmi\Pi}=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\omega_{1}}$ introduced by Definition 4.4.

Definition 5.1** (in ${\mathbf{L}}$).**

Define the multiforcings

[TABLE]

We further define $\boldsymbol{\mathbb{P}}=\text{MT}(\nmi\Pi)=\text{MT}(\vv{\nmi\Pi})$, and, for all $\gamma<\omega_{1}$,

[TABLE]

The multiforcing $\boldsymbol{\mathbb{P}}$ will be our principal forcing notion, the key forcing.

Lemma 5.2** (in ${\mathbf{L}}$).**

$\nmi\Pi$* is a regular multiforcing. In addition, $|\nmi\Pi|=\omega_{1}\times\omega$, thus if $\xi<\omega_{1}$ and $k<\omega$ then there is an ordinal $\alpha<\omega_{1}$ such that $\langle\xi,k\rangle\in|\nmi\Pi_{\alpha}|$. Therefore $\boldsymbol{\mathbb{P}}=\prod_{\xi<\omega_{1},\,k<\omega}\nmi\Pi(\xi,k)$ (with finite support).*

Proof**.**

To prove the additional claim, note that the set $W$ of all multisequences $\vv\boldsymbol{\pi}\in\vv\text{sMF}$ satisfying $\langle\xi,k\rangle\in|\bigcup^{\text{\tt cw}}\vv\boldsymbol{\pi}|$ is ${\mathbf{\Sigma}}^{\mathrm{HC}}_{1}$ (with $\xi$ as a parameter of definition). In addition $W$ is dense in $\vv\text{sMF}$. (First extend $\vv\boldsymbol{\pi}$ by Corollary 1.2 so that is has a non-limit length and the last term, then make use of Corollary 3.4.) Therefore by Lemma 4.5 there is an ordinal $\gamma<\omega_{1}$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\in W$, as required. ∎

If $\xi<\omega_{1}$ and $k<\omega$ then, following the lemma, let $\alpha(\xi,k)<\omega_{1}$ be the least ordinal $\alpha$ satisfying $\langle\xi,k\rangle\in|\nmi\Pi_{\alpha}|$. Thus a forcing $\nmi\Pi_{\alpha}(\xi,k)\in\text{{PTF}}$ is defined whenever $\alpha$ satisfies $\alpha(\xi,k)\leq\alpha<\omega_{1}$, and $\langle\nmi\Pi_{\alpha}(\xi,k)\rangle_{\alpha(\xi,k)\leq\alpha<\omega_{1}}$ is a ${\mathrel{\boldsymbol{\sqsubset}}}$-increasing sequence of countable special forcings in PTF.

Note that $\nmi\Pi(\xi,k)=\bigcup_{\alpha(\xi,k)\leq\alpha<\omega_{1}}\nmi\Pi_{\alpha}(\xi,k)$ by construction.

Corollary 5.3** (in ${\mathbf{L}}$).**

If $k<\omega$ then the sequence of ordinals $\langle\alpha(\xi,k)\rangle_{\xi<\omega_{1}}$ and the sequence of multiforcings $\langle\nmi\Pi_{\alpha}(\xi,k)\rangle_{\xi<\omega_{1},\,\alpha(\xi,k)\leq\alpha<\omega_{1}}$ are $\varDelta^{\mathrm{HC}}_{k+2}$.

Proof**.**

By construction the following double equivalence holds:

[TABLE]

However $\boldsymbol{\pi}=\nmi\Pi_{\alpha}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptk}$ is a $\varDelta^{\mathrm{HC}}_{k+2}$ relation by Theorem 4.1(i). It follows that so is the sequence $\langle\alpha(\xi,k)\rangle_{\xi<\omega_{1}}$. The second claim easily follows by the same Definition 4.4(i). ∎

Corollary 5.4** (in ${\mathbf{L}}$, of Lemma 1.2(v)).**

If $\xi<\omega_{1}$, $k<\omega$, and $\alpha(\xi,k)\leq\alpha<\omega_{1}$ then the set $\nmi\Pi_{\alpha}(\xi,k)$ is pre-dense in $\nmi\Pi(\xi,k)$ and in $\nmi\Pi$.∎

In spite of Lemma 5.2, the sets $|\nmi\Pi_{<\gamma}|$ can be quite arbitrary (countable) subsets of $\omega_{1}\times\omega$. However we easily get the next corollary:

Corollary 5.5** (in ${\mathbf{L}}$, of Lemma 5.2).**

The set ${{\mathbb{C}}}^{\prime}=\{\hskip 0.43057pt{\gamma<\omega_{1}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt|\nmi\Pi_{<\gamma}|=\gamma\times\omega\hskip 0.43057pt\}$ is a club in $\omega_{1}$.∎

Lemma 5.6** (in ${\mathbf{L}}$).**

$\boldsymbol{\mathbb{P}}$* is CCC.*

Proof**.**

Let $A\subseteq\boldsymbol{\mathbb{P}}$ be a maximal antichain in $\boldsymbol{\mathbb{P}}$. The set

[TABLE]

is a club in $\omega_{1}$. Let $D=\{\hskip 0.43057pt{{\boldsymbol{p}}\in\boldsymbol{\mathbb{P}}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\exists\,}{\boldsymbol{q}}\in A\,(p\leqslant q)\hskip 0.43057pt\}$; this is an open dense set. By Definition 4.4(iii), there is an ordinal $\gamma\in C$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset_{\{\hskip 0.3014ptD\cap\boldsymbol{\mathbb{P}}_{<\gamma}\hskip 0.3014pt\}}\vv{\nmi\Pi}$. Recall that $\gamma\in C$, hence $A\cap\boldsymbol{\mathbb{P}}_{<\gamma}$ is a maximal antichain in $\boldsymbol{\mathbb{P}}_{<\gamma}$, thus $D\cap\boldsymbol{\mathbb{P}}_{<\gamma}$ is open dense in $\boldsymbol{\mathbb{P}}_{<\gamma}$. Therefore the set $D\cap\boldsymbol{\mathbb{P}}_{<\gamma}$ is pre-dense in the forcing $\text{MT}(\vv{\nmi\Pi})=\boldsymbol{\mathbb{P}}$ by Corollary 1.2(v)(c). We claim that $A=A\cap\boldsymbol{\mathbb{P}}_{<\gamma}$, so $A$ is countable.

Indeed suppose that ${\boldsymbol{r}}\in A\smallsetminus\boldsymbol{\mathbb{P}}_{<\gamma}$. Then ${\boldsymbol{r}}$ is compatible with some ${\boldsymbol{q}}\in D\cap\boldsymbol{\mathbb{P}}_{<\gamma}$; let ${\boldsymbol{p}}\in D\cap\boldsymbol{\mathbb{P}}_{<\gamma}$, ${\boldsymbol{p}}\leqslant{\boldsymbol{q}}$, ${\boldsymbol{p}}\leqslant{\boldsymbol{r}}$. As ${\boldsymbol{q}}\in D$, there is some ${\boldsymbol{r}}^{\prime}\in A$ with ${\boldsymbol{q}}\leqslant{\boldsymbol{r}}^{\prime}$. Then ${\boldsymbol{r}}={\boldsymbol{r}}^{\prime}$ as $A$ is an antichain; thus ${\boldsymbol{q}}\leqslant{\boldsymbol{r}}\in A\smallsetminus\boldsymbol{\mathbb{P}}_{<\gamma}$. However ${\boldsymbol{q}}\in\boldsymbol{\mathbb{P}}_{<\gamma}$ and $A\cap\boldsymbol{\mathbb{P}}_{<\gamma}$ is a maximal antichain in $\boldsymbol{\mathbb{P}}_{<\gamma}$, thus ${\boldsymbol{q}}$, and hence ${\boldsymbol{r}}$, is compatible with some ${\boldsymbol{r}}^{\prime\prime}\in A\cap\boldsymbol{\mathbb{P}}_{<\gamma}$. Which is a contradiction. ∎

Corollary 5.7** (in ${\mathbf{L}}$).**

If a set $D\subseteq\boldsymbol{\mathbb{P}}$ is pre-dense in $\boldsymbol{\mathbb{P}}$ then there is an ordinal $\gamma<\omega_{1}$ such that $D\cap\boldsymbol{\mathbb{P}}_{<\gamma}$ is already pre-dense in $\boldsymbol{\mathbb{P}}$.

Proof**.**

We can assume that in fact $D$ is dense. Let $A\subseteq D$ be a maximal antichain in $D$; then $A$ is a maximal antichain in $\boldsymbol{\mathbb{P}}$ because of the density of $D$. Then $A\subseteq\boldsymbol{\mathbb{P}}_{<\gamma}$ for some $\gamma<\omega_{1}$ by Lemma 5.6. But $A$ is pre-dense in $\boldsymbol{\mathbb{P}}$. ∎

V Auxiliary forcing relation

Recall that $\boldsymbol{\mathbb{P}}=\text{MT}(\nmi\Pi)$, the key forcing, is a product forcing notion defined (in ${\mathbf{L}}$) in Section 5. Its components $\nmi\Pi(\xi,k)$ have different complexity in $\mathrm{HC}$, depending on $k$ by Corollary 5.3, hence there is no way the forcing notion $\boldsymbol{\mathbb{P}}$ (or $\nmi\Pi$) as a whole is definable in $\mathrm{HC}$. Somewhat surprisingly, the ${\boldsymbol{\mathbb{P}}}$-forcing relation turns out to be definable in $\mathrm{HC}$ when restricted to analytic formulas of a certain level of complexity within the usual hierarchy. This will be established on the base of an auxiliary forcing relation.

1 Auxiliary forcing: preliminaries

We argue in ${\mathbf{L}}$. Consider the 2nd order arithmetic language, with variables $k,l,m,n,\dots$ of type [math] over $\omega$ and variables $a,b,x,y,\dots$ of type $1$ over $\omega^{\omega}$, whose atomic formulas are those of the form $x(k)=n$. Let $\mathscr{L}$ be the extension of this language, which allows to substitute free variables of type [math] with natural numbers (as usual) and free variables of type $1$ with small real names $\mathbf{c}\in{\mathbf{L}}$. By ${\mathscr{L}}$-formulas we understand formulas of this extended language.

We define natural classes $\mathscr{L}\varSigma^{1}_{n}$, $\mathscr{L}\varPi^{1}_{n}$ ($n\geq 1$) of ${\mathscr{L}}$-formulas. Let $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ be the closure of $\mathscr{L}\varSigma^{1}_{1}\cup\mathscr{L}\varPi^{1}_{1}$ under ${\neg}\hskip 0.86108pt,\linebreak[0]\hskip 0.86108pt{\land}\hskip 0.86108pt,\linebreak[0]\hskip 0.86108pt{\lor}$ and quantifiers over $\omega$. If $\varphi$ is a formula in $\mathscr{L}\varSigma^{1}_{n}$ (resp., $\mathscr{L}\varPi^{1}_{n}$), then let $\varphi^{-}$ be the result of canonical transformation of $\neg\>\varphi$ to the $\mathscr{L}\varPi^{1}_{n}$ (resp., $\mathscr{L}\varSigma^{1}_{n}$) form.

If $\varphi$ is a ${\mathscr{L}}$-formula and $G\subseteq\text{MT}$ is a pairwise compatible set of multitrees then let $\varphi[G]$ be the result of substitution of $\mathbf{c}[G]$ for any name $\mathbf{c}$ in $\varphi$. (Recall Definition 1.2.) Thus $\varphi[G]$ is an ordinary 2nd order arithmetic formula, which may include natural numbers and elements of $\omega^{\omega}$ as parameters.

We are going to define a relation ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ between multitrees ${\boldsymbol{p}}$, multisequences $\vv\boldsymbol{\pi}$, and ${\mathscr{L}}$-formulas $\varphi$, which suitably approximates the true ${\boldsymbol{\mathbb{P}}}$-forcing relation. But it depends on a two more definitions.

Definition 1.1**.**

If $m<\omega$ then $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ consists of all multisequences $\vv\boldsymbol{\pi}\in\vv\text{sMF}$ such that ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}\subset{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, that is, ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}=({\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}){\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\delta$, where $\delta=\mathop{\tt dom}(\vv\boldsymbol{\pi})$ — multisequences which agree with the key multisequence $\vv{\nmi\Pi}$ on layers below $m$. Obviously $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm+1}]\subseteq\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]\subseteq\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275pt0}]=\vv\text{sMF}$. ∎

If $\gamma<\omega_{1}$ then the subsequence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ of the key multisequence $\vv{\nmi\Pi}$ belongs to $\bigcap_{m}\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$, of course. To prove the next lemma use 4.4(i).

Lemma 1.2**.**

$\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$* is a subset of $\mathrm{HC}$ of definability class $\varDelta^{\mathrm{HC}}_{m+1}$.∎*

The other definition deals with models of a subtheory of ZFC.

Definition 1.3**.**

Let $\text{ZFL}^{\text{--}}$ be the theory containing all axioms of $\text{ZFC}^{-}$ (minus the Power Set axiom) plus the axiom of constructibility ${\mathbf{V}}={\mathbf{L}}$. Any transitive model (TM) of $\text{ZFL}^{\text{--}}$ has the form ${\mathbf{L}}_{\alpha}$, where $\alpha\in\text{Ord}$. Therefore it is true in ${\mathbf{L}}$ that for any set $x$ there is a least TM ${{\mathfrak{M}}}={{\mathfrak{M}}}(x)$ of $\text{ZFL}^{\text{--}}$ containing $x$. If $x\in\mathrm{HC}$ ($\mathrm{HC}$= all hereditarily countable sets) then ${{\mathfrak{M}}}(x)$ is a countable transitive model (CTM), equal to the least CTM of $\text{ZFL}^{\text{--}}$ containing $x$. ∎

2 Auxiliary forcing

The definition of ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ goes on by induction on the complexity of $\varphi$.

1. $1^{\circ}$.

Let $\varphi$ is a closed $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ formula, $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, ${\boldsymbol{p}}\in\text{MT}$, but ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$ is not necessarily assumed. We define:

1. (a)

${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ iff there is a CTM ${{\mathfrak{M}}}\models\text{ZFL}^{\text{--}}$, an ordinal $\vartheta<\mathop{\tt dom}\vv\boldsymbol{\pi}$, and a multitree ${\boldsymbol{p}}_{0}\in\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$, such that ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}$ (meaning: ${\boldsymbol{p}}$ is stronger), the model ${{\mathfrak{M}}}$ contains $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta$ (then contains $\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$ as well) and contains $\varphi$ (that is, all names in $\varphi$), $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta\subset_{{\mathfrak{M}}}\vv\boldsymbol{\pi}$, and ${\boldsymbol{p}}_{0}$ ${\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)}$-forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}$ (in the usual sense) ; 2. (b)

${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ (weak forcing) iff there is no multisequence $\vv\boldsymbol{\pi}^{\prime}\in\vv\text{sMF}$ and ${\boldsymbol{p}}^{\prime}\in\text{MT}(\vv\boldsymbol{\pi}^{\prime})$ such that $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\neg\>\varphi$. 2. $2^{\circ}$.

If $\varphi(x)$ is a $\mathscr{L}\varPi^{1}_{n}$ formula, $n\geq 1$, then we define ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}{\exists\,}x\,\varphi(x)$ iff there is a small real name $\mathbf{c}$ such that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi(\mathbf{c})$. 3. $3^{\circ}$.

If $\varphi$ is a closed $\mathscr{L}\varPi^{1}_{n}$ formula, $n\geq 2$, then we define ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ iff $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, and there is no multisequence $\vv\boldsymbol{\pi}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ and multitree ${\boldsymbol{p}}^{\prime}\in\text{MT}(\vv\boldsymbol{\pi}^{\prime})$ such that $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\varphi^{-}$. 555 If $\vv\boldsymbol{\pi}$ does not belong to $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ in $3^{\circ}$, then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ holds for any $\mathscr{L}\varPi^{1}_{n}$ formula $\varphi$ by default as a multisequence not in $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ is definitely not extendable to a multisequence in $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$. This motivates the condition $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ in $3^{\circ}$.

Remark 2.1**.**

With ${\boldsymbol{p}}_{0}$ and $\vartheta$ given, the premise “${\boldsymbol{p}}_{0}$ ${\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)}$-forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}$” of $1^{\circ}$a does not depend on the choice of a CTM ${{\mathfrak{M}}}$ containing $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta$ and $\varphi$, since if $\varphi$ is $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ then the formula $\varphi[G]$ (in which all names are evaluated by some ${\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)}$-generic set $G$ as in 1.2) in simultaneously true or false in all transitive models by the Shoenfield absoluteness theorem. ∎

Remark 2.2**.**

It easily holds by induction that if ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ then $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, $\varphi$ is a closed formula in one of the classes $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1},\,\mathscr{L}\varSigma^{1}_{n},\,\mathscr{L}\varPi^{1}_{n}$, $n\geq 2$, and if $n\geq 2$ and $\varphi\in\mathscr{L}\varPi^{1}_{n}\cup\mathscr{L}\varSigma^{1}_{n+1}$ then $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$. ∎

Lemma 2.3**.**

Assume that multisequences $\vv\boldsymbol{\pi}\subseteq\vv\text{\leavevmode{}}$ belong to $\vv\text{sMF}$, ${\boldsymbol{q}},{\boldsymbol{p}}\in\text{MT}$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, $\varphi$ is an ${\mathscr{L}}$-formula as in 2.2, and if $n\geq 2$ and $\varphi\in\mathscr{L}\varPi^{1}_{n}\cup\mathscr{L}\varSigma^{1}_{n+1}$ then $\vv\boldsymbol{\pi},\vv\text{\leavevmode{}}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$. Then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ implies ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$, and if $\varphi$ belongs to $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ implies ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$ as well.

Proof**.**

If $\varphi$ is a $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ formula, ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, and this is witnessed by ${{\mathfrak{M}}}$, $\vartheta$, ${\boldsymbol{p}}_{0}$ as in $1^{\circ}$a, then the exactly same ${{\mathfrak{M}}}$, $\vartheta$, ${\boldsymbol{p}}_{0}$ witness ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$.

The induction step ${\exists\,}$, as in $2^{\circ}$, is elementary.

Now the induction step ${\forall\,}$, as in $3^{\circ}$. Let $\varphi$ be a closed ${\mathscr{L}\varPi^{1}_{n}}$-formula, $n\geq 2$, and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$. Assume to the contrary that ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$ fails. Then by $3^{\circ}$ there exist: a multisequence $\vv\text{\leavevmode{}}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ and multitree ${\boldsymbol{q}}^{\prime}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ such that $\vv\text{\leavevmode{}}\subseteq\vv\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{q}}^{\prime}\leqslant{\boldsymbol{q}}$, and ${\boldsymbol{q}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\varphi^{-}$. But then $\vv\boldsymbol{\pi}\subseteq\vv\text{\leavevmode{}}^{\prime}$ and ${\boldsymbol{q}}^{\prime}\leqslant{\boldsymbol{p}}$, hence ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ fails by $3^{\circ}$, which is a contradiction.

The additional result for $\mathop{\hskip 1.72218pt{\tt wforc}\hskip 1.72218pt}$ and $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ formulas is entirely similar to the induction step ${\forall\,}$ as just above. ∎

If $K$ is one of the classes $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$, $\mathscr{L}\varSigma^{1}_{n}$, $\mathscr{L}\varPi^{1}_{n}$ ($n\geq 2$), then let $\text{FORC}[K]$ consist of all triples $\langle\vv\boldsymbol{\pi},{\boldsymbol{p}},\varphi\rangle$ such that $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$, $\varphi$ is a closed ${\mathscr{L}}$-formula of class $K$, and if $n\geq 2$ and $\varphi\in\mathscr{L}\varSigma^{1}_{n}\cup\mathscr{L}\varPi^{1}_{n}$ then $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, and finally ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$. Then $\text{FORC}[K]$ is a subset of $\mathrm{HC}$.

Lemma 2.4** (definability, in ${\mathbf{L}}$).**

$\text{FORC}[\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}]$* belongs to $\varDelta^{\mathrm{HC}}_{1}$. If $n\geq 2$ then $\text{FORC}[\mathscr{L}\varSigma^{1}_{n}]$ belongs to $\varSigma^{\mathrm{HC}}_{n-1}$ and $\text{FORC}[\mathscr{L}\varPi^{1}_{n}]$ belongs to $\varPi^{\mathrm{HC}}_{n-1}$.*

Proof**.**

Relations like “being an MSP”, “being a formula in $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$, $\mathscr{L}\varSigma^{1}_{n}$, $\mathscr{L}\varPi^{1}_{n}$”, ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\rho})$, forcing over a CTM, etc. are definable in $\mathrm{HC}$ by bounded formulas, hence $\varDelta^{\mathrm{HC}}_{1}$. On the top of this, the model ${{\mathfrak{M}}}$ can be tied by both ${\exists\,}$ and ${\forall\,}$ in $1^{\circ}$a, see Remark 2.1. This wraps up the $\varDelta^{\mathrm{HC}}_{1}$ estimation for $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$.

The inductive step by $2^{\circ}$ is quite simple.

Now the step by $3^{\circ}$. Assume that $n\geq 2$, and it is already established that $\text{FORC}[\mathscr{L}\varSigma^{1}_{n}]\in\varSigma^{\mathrm{HC}}_{n-1}$. Then $\langle\vv\boldsymbol{\pi},{\boldsymbol{p}},\varphi\rangle\in\text{FORC}[\mathscr{L}\varPi^{1}_{n}]$ iff $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, ${\boldsymbol{p}}\in\text{MT}$, $\varphi$ is a closed $\mathscr{L}\varPi^{1}_{n}$ formula in ${{\mathfrak{M}}}(\vv\boldsymbol{\pi})$, and, by $3^{\circ}$, there exist no triple $\langle\vv\boldsymbol{\pi}^{\prime},{\boldsymbol{p}}^{\prime},\psi\rangle\in\text{FORC}[\mathscr{L}\varSigma^{1}_{n}]$ such that $\vv\boldsymbol{\pi}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, ${\boldsymbol{p}}^{\prime}\in\text{MT}(\vv\boldsymbol{\pi}^{\prime})$, ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, and $\psi$ is $\varphi^{-}$. Evaluating the key conjunct $\vv\boldsymbol{\pi}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ by Lemma 1.2 as $\varDelta^{\mathrm{HC}}_{n-1}$, we get a required estimation $\varPi^{\mathrm{HC}}_{n-1}$ of $\text{FORC}[\mathscr{L}\varPi^{1}_{n}]$. ∎

3 Forcing simple formulas

We still argue in ${\mathbf{L}}$. The following results are mainly related to the relation $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$ with respect to formulas in the class $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$.

Lemma 3.1** (in ${\mathbf{L}}$).**

*Assume that $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, $\vv\text{\leavevmode{}}\in\vv\text{sMF}\cup\vv\text{sMF}_{\omega_{1}}$, $\vv\boldsymbol{\pi}\subseteq\vv\text{\leavevmode{}}$, ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$, $\varphi$ is a formula in $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$, ${{\mathfrak{N}}}\models\text{ZFL}^{\text{--}}$ is a TM containing $\vv\text{\leavevmode{}}$ and $\varphi$, and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$. Then ${\boldsymbol{p}}$ ${\text{MT}(\vv\text{\leavevmode{}})}$-forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{N}}}$. *

Proof**.**

By definition there is an ordinal $\vartheta<\mathop{\tt dom}\vv\boldsymbol{\pi}$, a multitree ${\boldsymbol{p}}_{0}\in\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$, and a CTM ${{\mathfrak{M}}}\models\text{ZFL}^{\text{--}}$ containing $\varphi$ and $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta$, such that $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta\subset_{{\mathfrak{M}}}\vv\boldsymbol{\pi}$, ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}$, and ${\boldsymbol{p}}_{0}$ ${\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)}$-forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}$. We can w. l. o. g. assume that ${{\mathfrak{M}}}\subseteq{{\mathfrak{N}}}$ (by the same reference to Shoenfield as in Remark 2.1).

Now suppose that $G\subseteq\text{MT}(\vv\text{\leavevmode{}})$ is a set ${\text{MT}(\vv\text{\leavevmode{}})}$-generic over ${{\mathfrak{N}}}$ and ${\boldsymbol{p}}\in G$ — then ${\boldsymbol{p}}_{0}\in G$, too. We have to prove that $\varphi[G]$ is true in ${{\mathfrak{N}}}[G]$.

We claim that the set $G^{\prime}=G\cap\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$ is ${\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)}$-generic over ${{\mathfrak{M}}}$. Indeed, let a set ${\boldsymbol{D}}\in{{\mathfrak{M}}}$, ${\boldsymbol{D}}\subseteq\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$, be open dense in $\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$. Then, as ${\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$ by Corollary 1.2(iv), ${\boldsymbol{D}}$ is pre-dense in $\text{MT}(\vv\text{\leavevmode{}})$ by 1.2(v)(c), and hence $G\cap{\boldsymbol{D}}\neq\varnothing$ by the choice of $G$. It follows that $G^{\prime}\cap{\boldsymbol{D}}\neq\varnothing$.

We claim that $\mathbf{c}[G]=\mathbf{c}[G^{\prime}]$ for any name $\mathbf{c}\in{{\mathfrak{M}}}$, in particular, for any name in $\varphi$. Indeed, as $G^{\prime}\subseteq G$, the otherwise occurs by Definition 1.2 only if for some $n,i$ and ${\boldsymbol{q}}^{\prime}\in K^{\mathbf{c}}_{ni}$ there is ${\boldsymbol{q}}\in G$ satisfying ${\boldsymbol{q}}\leqslant{\boldsymbol{q}}^{\prime}$, but there is no such ${\boldsymbol{q}}$ in $G^{\prime}$. Let ${\boldsymbol{D}}$ consist of all multitrees ${\boldsymbol{r}}\in\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$ either satisfying ${\boldsymbol{r}}\leqslant{\boldsymbol{q}}^{\prime}$ or somewhere a. d. with ${\boldsymbol{q}}^{\prime}$. Then ${\boldsymbol{D}}\in{{\mathfrak{M}}}$ and ${\boldsymbol{D}}$ is open dense in $\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$. Therefore ${\boldsymbol{D}}\cap G^{\prime}\neq\varnothing$ by the above, so let ${\boldsymbol{r}}\in{\boldsymbol{D}}\cap G^{\prime}$. If ${\boldsymbol{r}}\leqslant{\boldsymbol{q}}^{\prime}$ then we get a contradiction with the choice of ${\boldsymbol{q}}^{\prime}$. If ${\boldsymbol{r}}$ is somewhere a. d.  with ${\boldsymbol{q}}^{\prime}$ then we get a contradiction with the choice of ${\boldsymbol{q}}$ as both ${\boldsymbol{q}},{\boldsymbol{r}}$ belong to the generic filter $G$.

It follows that $\varphi[G]$ coincides with $\varphi[G^{\prime}]$.

Note also that ${\boldsymbol{p}}_{0}\in G^{\prime}$. We conclude that $\varphi[G^{\prime}]$ is true in ${{\mathfrak{M}}}[G^{\prime}]$ as ${\boldsymbol{p}}_{0}$ forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}$. The same formula $\varphi[G]$ is true in ${{\mathfrak{N}}}[G]$ by Shoenfield. ∎

Lemma 3.2**.**

Let $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$, $\varphi$ be a formula in $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$. Then

1. (i)

${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$* and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\neg\>\varphi$ cannot hold together**;*** 2. (ii)

if ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi\;;$ 3. (iii)

if ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ then there is a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ such that $\vv\boldsymbol{\pi}\subset_{{{\mathfrak{M}}}(\vv\boldsymbol{\pi})}\vv\text{\leavevmode{}}$ and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$; 4. (iv)

${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$* and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\neg\>\varphi$ cannot hold together.*

Proof**.**

(i) Otherwise ${\boldsymbol{p}}$ ${\text{MT}(\vv\boldsymbol{\pi})}$-forces both $\varphi[{\underline{G}}]$ and $\neg\>\varphi[{\underline{G}}]$ over a large enough CTM ${{\mathfrak{M}}}$, by Lemma 3.1, which cannot happen.

(ii) Assume that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ fails, hence there is a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ and a multitree ${\boldsymbol{q}}\in\text{MT}(\vv\text{\leavevmode{}})$ such that ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$ and ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\neg\>\varphi$. But Lemma 2.3 implies ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, which contradicts to (i).

(iii) Let ${{\mathfrak{M}}}\models\text{ZFL}^{\text{--}}$ be a CTM containing $\varphi$ and $\vv\boldsymbol{\pi}$. By Corollary 1.2(ii), there is a multisequence multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ with $\vv\boldsymbol{\pi}\subset_{{{\mathfrak{M}}}}\vv\text{\leavevmode{}}$. We claim that ${\boldsymbol{p}}\,$ ${\text{MT}(\vv\boldsymbol{\pi})}$-forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}$ in the usual sense — then by definition ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$ (via $\vartheta=\mathop{\tt dom}\vv\boldsymbol{\pi}$), and we are done. To prove the claim suppose otherwise. Then there is a multitree ${\boldsymbol{q}}\in\text{MT}(\vv\boldsymbol{\pi})$ such that ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$ and ${\boldsymbol{q}}$ ${\text{MT}(\vv\boldsymbol{\pi})}$-forces $\neg\>\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}$, thus ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\neg\,\varphi$. But this contradicts to ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$.

(iv) There is a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ by (iii), such that $\vv\boldsymbol{\pi}\subset\vv\text{\leavevmode{}}$ and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$. Note that still ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\neg\>\varphi$ by Lemma 2.3. Extend $\vv\text{\leavevmode{}}$ once again, getting a contradiction with (i). ∎

Corollary 3.3**.**

Let $n\geq 2$, $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$, $\varphi$ be a formula in $\mathscr{L}\varSigma^{1}_{n}$. Then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi^{-}$ cannot hold together.

Proof**.**

If $n=1$ then apply Lemma 3.2(i). If $n\geq 2$ then the result immediately follows by definition ($3^{\circ}$ in Section 2). ∎

Corollary 3.4** (in ${\mathbf{L}}$).**

Assume that $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$, $\varphi$ is a formula in $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$, ${{\mathfrak{N}}}\models\text{ZFL}^{\text{--}}$ is a TM containing $\vv\boldsymbol{\pi}$ and $\varphi$, and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$. Then ${\boldsymbol{p}}$ ${\text{MT}(\vv\boldsymbol{\pi})}$-forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{N}}}$ in the usual sense.

This looks like the case $\vv\boldsymbol{\rho}=\vv\boldsymbol{\pi}$ in Lemma 3.1, but $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$ is weakened to $\mathop{\hskip 1.72218pt{\tt wforc}\hskip 1.72218pt}$, and $\varphi\in{{\mathfrak{M}}}$ (automatic in Lemma 3.1) is added, in the premise.

Proof**.**

Otherwise there is a multitree ${\boldsymbol{q}}\in\text{MT}(\vv\boldsymbol{\pi})$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, that ${\text{MT}(\vv\boldsymbol{\pi})}$-forces $\neg\>\varphi[{\underline{G}}]$ over ${{\mathfrak{N}}}$. On the other hand, by Lemma 3.2(iii), there is a multisequence $\vv\text{\leavevmode{}}\in\vv\text{sMF}$ such that $\vv\boldsymbol{\pi}\subset_{{{\mathfrak{M}}}(\vv\boldsymbol{\pi})}\vv\text{\leavevmode{}}$ and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$, hence, ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$ by Lemma 2.3. However we have ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\neg\,\varphi$ by definition ($1^{\circ}$a in Section 2 with $\vartheta=\mathop{\tt dom}\vv\boldsymbol{\pi}$), which contradicts to Lemma 3.2(i). ∎

4 Tail invariance

If $\vv\boldsymbol{\pi}=\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}\in\vv\text{sMF}$ and $\gamma<\lambda=\mathop{\tt dom}\vv\boldsymbol{\pi}$ then let the ${\gamma}$-tail $\vv\boldsymbol{\pi}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma}$ be the restriction $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{[\gamma,\lambda)}$ to the ordinal semiinterval $[\gamma,\lambda)=\{\hskip 0.43057pt{\alpha\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\gamma\leq\alpha<\lambda\hskip 0.43057pt\}$. Then the multiforcing $\text{MT}(\vv\boldsymbol{\pi}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma})=\bigcup^{\text{\tt cw}}_{\gamma\leq\alpha<\lambda}\vv\boldsymbol{\pi}(\alpha)$ is open dense in $\text{MT}(\vv\boldsymbol{\pi})$ by Corollary 1.2(v)(a). Therefore it can be expected that if $\vv\text{\leavevmode{}}$ is another multisequence of the same length $\lambda=\mathop{\tt dom}\vv\text{\leavevmode{}}$, and $\vv\text{\leavevmode{}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma}=\vv\boldsymbol{\pi}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma}$, then the relation $\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}$ coincides with $\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}$. And indeed this turns out to be the case (almost).

Theorem 4.1**.**

Assume that $\vv\boldsymbol{\pi},\vv\text{\leavevmode{}}$ are multisequences in $\vv\text{sMF}$, $\gamma<\lambda=\mathop{\tt dom}\vv\boldsymbol{\pi}=\mathop{\tt dom}\vv\text{\leavevmode{}}$, $\vv\text{\leavevmode{}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma}=\vv\boldsymbol{\pi}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma}$, ${\boldsymbol{p}}\in\text{MT}$, and $\varphi$ is an ${\mathscr{L}}$-formula. Then

1. (i)

if $\varphi\in\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ iff ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi\;;$ 2. (ii)

if $n\geq 2$, $\vv\boldsymbol{\pi},\vv\text{\leavevmode{}}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, and $\varphi\in\mathscr{L}\varPi^{1}_{n}\cup\mathscr{L}\varSigma^{1}_{n+1}$, then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ iff ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$.

Proof**.**

(i) Suppose to the contrary that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, but ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$ fails, so there is a multisequence $\vv\text{\leavevmode{}}^{\prime}\in\vv\text{sMF}$ and ${\boldsymbol{p}}^{\prime}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ such that $\vv\text{\leavevmode{}}\subset\vv\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\neg\>\varphi$. Let $\lambda^{\prime}=\mathop{\tt dom}\vv\text{\leavevmode{}}^{\prime}$. By Corollary 1.2(v)(a), there is a multitree ${\boldsymbol{r}}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma})$, ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}^{\prime}$. Then still ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}$ and ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\neg\>\varphi$, by Lemma 2.3.

Define a multisequence $\vv\boldsymbol{\pi}^{\prime}$ so that $\mathop{\tt dom}{\vv\boldsymbol{\pi}^{\prime}}=\lambda^{\prime}=\mathop{\tt dom}{\vv\text{\leavevmode{}}^{\prime}}$, $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, and $\vv\boldsymbol{\pi}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}=\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}$. Then ${\boldsymbol{r}}\in\text{MT}(\vv\boldsymbol{\pi}^{\prime})$, and ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\varphi$ by Lemma 2.3.

Consider any CTM ${{\mathfrak{N}}}\models\text{ZFL}^{\text{--}}$ containing $\varphi$, $\vv\boldsymbol{\pi}^{\prime}$, $\vv\text{\leavevmode{}}^{\prime}$. Then, by Corollary 3.1, one and the same multitree ${\boldsymbol{r}}$ ${\text{MT}(\vv\boldsymbol{\pi}^{\prime})}$-forces $\varphi[{\underline{G}}]$ but ${\text{MT}(\vv\text{\leavevmode{}}^{\prime})}$-forces $\neg\>\varphi[{\underline{G}}]$ over ${{\mathfrak{N}}}$. But this contradicts to the fact that the forcing notions $\text{MT}(\vv\boldsymbol{\pi}^{\prime})$, $\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ contain one and the same dense set $\text{MT}(\vv\boldsymbol{\pi}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda})=\text{MT}(\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda})$.

(ii) Consider first the $\mathscr{L}\varPi^{1}_{2}$ case. Assume that $\varphi(x)$ is a $\mathscr{L}\varSigma^{1}_{1}$ formula, ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}{\forall\,}x\,\varphi(x)$, but to the contrary ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}{\forall\,}x\,\varphi(x)$ fails. Thus there is a multisequence $\vv\text{\leavevmode{}}^{\prime}\in\vv\text{sMF}$ and a multitree ${\boldsymbol{p}}^{\prime}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ such that $\vv\text{\leavevmode{}}\subseteq\vv\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}{\exists\,}x\,\varphi^{-}(x)$. By definition there is a small real name $\mathbf{c}$ such that ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\varphi^{-}(\mathbf{c})$. There is a multitree ${\boldsymbol{r}}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma})$, ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}^{\prime}$. Then still ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}$ and ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\varphi^{-}(\mathbf{c})$. As above there is a multisequence $\vv\boldsymbol{\pi}^{\prime}$ such that $\mathop{\tt dom}{\vv\boldsymbol{\pi}^{\prime}}=\lambda^{\prime}=\mathop{\tt dom}{\vv\text{\leavevmode{}}^{\prime}}$, $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, and $\vv\boldsymbol{\pi}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}=\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}$. Then ${\boldsymbol{r}}\in\text{MT}(\vv\boldsymbol{\pi}^{\prime})$ and ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\varphi^{-}(\mathbf{c})$ by the inductive hypothesis. By Lemma 3.2, there is a multisequence $\vv\boldsymbol{\sigma}$ such that $\vv\boldsymbol{\pi}^{\prime}\subseteq\vv\boldsymbol{\sigma}$ and ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\sigma}}\hskip 1.72218pt}\varphi^{-}(\mathbf{c})$, hence, ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\sigma}}\hskip 1.72218pt}{\exists\,}x\,\varphi^{-}(x)$. But this contradicts to ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}{\forall\,}x\,\varphi(x)$, since ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}$.

To carry out the step $\mathscr{L}\varPi^{1}_{n}\to\mathscr{L}\varSigma^{1}_{n+1}$, $n\geq 2$, let $\varphi(x)$ be a formula in $\mathscr{L}\varPi^{1}_{n}$. Assume that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}{\exists\,}x\,\varphi(x)$. By definition (see $2^{\circ}$ in Section 2), there is a small real name $\mathbf{c}$ such that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi(\mathbf{c})$. Then we have ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi(\mathbf{c})$ by the inductive assumption, thus ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}{\exists\,}x\,\psi(x)$.

To carry out the step $\mathscr{L}\varSigma^{1}_{n}\to\mathscr{L}\varPi^{1}_{n}$, $n\geq 3$, assume that $\varphi$ is a $\mathscr{L}\varPi^{1}_{n}$ formula, ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, but to the contrary ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$ fails. Then by $3^{\circ}$ of Section 2, as $\vv\text{\leavevmode{}}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, there is a multisequence $\vv\text{\leavevmode{}}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ and a multitree ${\boldsymbol{p}}^{\prime}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ such that $\vv\text{\leavevmode{}}\subseteq\vv\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\varphi^{-}$. By Corollary 1.2(v)(a), there is a multitree ${\boldsymbol{r}}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma})$, ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}^{\prime}$. Then still ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}$ and ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\varphi^{-}$. As above in the proof of (i), there is a multisequence $\vv\boldsymbol{\pi}^{\prime}$ such that $\mathop{\tt dom}{\vv\boldsymbol{\pi}^{\prime}}=\lambda^{\prime}=\mathop{\tt dom}{\vv\text{\leavevmode{}}^{\prime}}$, $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, and $\vv\boldsymbol{\pi}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}=\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}$.

We claim that $\vv\boldsymbol{\pi}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$. Indeed if $\alpha<\mathop{\tt dom}\vv\boldsymbol{\pi}$ then $\vv\boldsymbol{\pi}^{\prime}(\alpha)=\vv\boldsymbol{\pi}(\alpha)$ (as $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$), hence ${\vv\boldsymbol{\pi}^{\prime}(\alpha)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}={\vv\boldsymbol{\pi}(\alpha)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}={\nmi\Pi_{\alpha}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}$ (as $\vv\boldsymbol{\pi}$ belongs to $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$). If $\mathop{\tt dom}\vv\boldsymbol{\pi}\leq\alpha<\mathop{\tt dom}{\vv\boldsymbol{\pi}^{\prime}}$ then $\vv\boldsymbol{\pi}^{\prime}(\alpha)=\vv\text{\leavevmode{}}^{\prime}(\alpha)$ (as $\vv\boldsymbol{\pi}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}=\vv\text{\leavevmode{}}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\lambda}$), hence ${\vv\boldsymbol{\pi}^{\prime}(\alpha)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}={\vv\text{\leavevmode{}}^{\prime}(\alpha)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}={\nmi\Pi_{\alpha}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}$ (as $\vv\text{\leavevmode{}}^{\prime}$ belongs to $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$). Thus ${\vv\boldsymbol{\pi}^{\prime}(\alpha)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}={\nmi\Pi_{\alpha}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}$ for all $\alpha$, meaning that ${\vv\boldsymbol{\pi}^{\prime}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}\subset{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}$ and $\vv\boldsymbol{\pi}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$. To conclude, $\vv\boldsymbol{\pi}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, ${\boldsymbol{r}}\in\text{MT}(\vv\boldsymbol{\pi}^{\prime}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\gamma})$, ${\boldsymbol{r}}\leqslant{\boldsymbol{p}}$, and also ${\boldsymbol{r}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\varphi^{-}$ by the inductive hypothesis. But this contradicts to the assumption ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$. ∎

5 Permutations

Still arguing in ${\mathbf{L}}$, we let PERM be the set of all bijections $\boldsymbol{h}:\omega_{1}\times\omega\overset{{\text{onto}}}{\longrightarrow}\omega_{1}\times\omega$, such that the kernel $|\boldsymbol{h}|=\{\hskip 0.43057pt{\langle\xi,k\rangle\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\boldsymbol{h}(\xi,k)\neq\langle\xi,k\rangle\hskip 0.43057pt\}$ is at most countable. Elements of PERM will be called permutations. If $m<\omega$ then let $\text{\rm PERM}_{m}$ consist of those permutations $\boldsymbol{h}\in\text{\rm PERM}$ satisfying $|\boldsymbol{h}|\subseteq\omega_{1}\times(\omega_{1}\smallsetminus m)$. In other words, if $\boldsymbol{h}\in\text{\rm PERM}_{m}$ and $\xi<\omega_{1}$, $k<m$, then $\boldsymbol{h}(\xi,k)=\langle\xi,k\rangle$.

Let $\boldsymbol{h}\in\text{\rm PERM}$. We extend the action of $\boldsymbol{h}$ as follows.

• •

if ${\boldsymbol{p}}$ is a multitree then $\boldsymbol{h}{\boldsymbol{p}}$ is a multitree, $|\boldsymbol{h}{\boldsymbol{p}}|={\boldsymbol{h}}\text{\rm\hskip 1.29167pt''}{{\boldsymbol{p}}}=\{\hskip 0.43057pt{\boldsymbol{h}(\xi,k)\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\langle\xi,k\rangle\in|{\boldsymbol{p}}|\hskip 0.43057pt\}$, and $(\boldsymbol{h}{\boldsymbol{p}})(\boldsymbol{h}(\xi,k))={\boldsymbol{p}}(\xi,k)$ whenever $\langle\xi,k\rangle\in|{\boldsymbol{p}}|$, in other words, $\boldsymbol{h}{\boldsymbol{p}}$ coincides with the superposition ${\boldsymbol{p}}\circ{(\boldsymbol{h}^{-1})}$;

• •

if $\boldsymbol{\pi}\in\text{MT}$ is a multiforcing then $\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\boldsymbol{\pi}=\boldsymbol{\pi}\circ{(\boldsymbol{h}^{-1})}$ is a multiforcing, $|\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\boldsymbol{\pi}|={\boldsymbol{h}}\text{\rm\hskip 1.29167pt''}{\boldsymbol{\pi}}$ and $(\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\boldsymbol{\pi})(\boldsymbol{h}(\xi,k))=\boldsymbol{\pi}(\xi,k)$ whenever $\langle\xi,k\rangle\in|\boldsymbol{\pi}|$;

• •

if $\mathbf{c}\subseteq\text{MT}\times(\omega\times\omega)$ is a real name, then put $\boldsymbol{h}\mathbf{c}=\{\hskip 0.43057pt{\langle\boldsymbol{h}{\boldsymbol{p}},n,i\rangle\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\langle{\boldsymbol{p}},n,i\rangle\in\mathbf{c}\hskip 0.43057pt\}$, thus easily $\boldsymbol{h}\mathbf{c}$ is a real name as well;

• •

if $\vv\boldsymbol{\pi}=\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\kappa}$ is a multisequence, then $\boldsymbol{h}\vv\boldsymbol{\pi}=\langle\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\kappa}$, still a multisequence.

• •

if $\varphi:=\varphi(\mathbf{c}_{1},\dots,\mathbf{c}_{n})$ is a ${\mathscr{L}}$-formula (with all names explicitly indicated), then $\boldsymbol{h}\varphi$ is $\varphi(\boldsymbol{h}\mathbf{c}_{1},\dots,\boldsymbol{h}\mathbf{c}_{n})$.

Many notions and relations defined above are clearly PERM-invariant, e. g., ${\boldsymbol{p}}\in\text{MT}(\boldsymbol{\pi})$ iff $\boldsymbol{h}{\boldsymbol{p}}\in\text{MT}(\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\boldsymbol{\pi})$, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$ iff ${\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\boldsymbol{\pi}}\mathrel{\boldsymbol{\sqsubset}}{\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\text{\leavevmode{}}}$, et cetera. The invariance persists even with respect to the relation $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$, at least to some extent.

Theorem 5.1**.**

*Assume that $\vv\boldsymbol{\pi}\in\vv\text{sMF}$, ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$, $\varphi$ is an ${\mathscr{L}}$-formula, and $\boldsymbol{h}\in\text{\rm PERM}$. Then *

1. (i)

if $\varphi$ belongs to $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$ and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, then $(\boldsymbol{h}{\boldsymbol{p}})\mathop{\hskip 1.72218pt{\tt wforc}_{\boldsymbol{h}\vv\boldsymbol{\pi}}\hskip 1.72218pt}(\boldsymbol{h}\varphi)\;;$ 2. (ii)

if $n\geq 2$, $\boldsymbol{h}\in\text{\rm PERM}_{n-2}$, and $\varphi$ belongs to $\mathscr{L}\varPi^{1}_{n}\cup\mathscr{L}\varSigma^{1}_{n+1}$, then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ iff $(\boldsymbol{h}{\boldsymbol{p}})\mathop{\hskip 1.72218pt{\tt forc}_{\boldsymbol{h}\vv\boldsymbol{\pi}}\hskip 1.72218pt}(\boldsymbol{h}\varphi)$.

Proof**.**

Let $\vv\text{\leavevmode{}}=\boldsymbol{h}\vv\boldsymbol{\pi}$, ${\boldsymbol{q}}=\boldsymbol{h}{\boldsymbol{p}}$, $\psi:=\boldsymbol{h}\varphi$.

(i) Suppose to the contrary that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, but ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\psi$ fails, so that there is a multisequence $\vv\text{\leavevmode{}}^{\prime}\in\vv\text{sMF}$ and ${\boldsymbol{q}}^{\prime}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ such that $\vv\text{\leavevmode{}}\subset\vv\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{q}}^{\prime}\leqslant{\boldsymbol{q}}$, and ${\boldsymbol{q}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\neg\>\psi$. The multisequence $\vv\boldsymbol{\pi}^{\prime}=\boldsymbol{h}^{-1}\vv\text{\leavevmode{}}^{\prime}$ then satisfies $\vv\boldsymbol{\pi}\subset\vv\boldsymbol{\pi}^{\prime}$, and the multitree ${\boldsymbol{p}}^{\prime}=\boldsymbol{h}^{-1}{\boldsymbol{q}}^{\prime}$ belongs to $\text{MT}(\vv\boldsymbol{\pi}^{\prime})$ and ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, hence we have ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\varphi$ by Lemma 2.3.

Now let ${{\mathfrak{M}}}\models\text{ZFL}^{\text{--}}$ be an arbitrary CTM containing $\vv\boldsymbol{\pi}^{\prime},\vv\text{\leavevmode{}}^{\prime},\varphi,\psi,\boldsymbol{h}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}{|\boldsymbol{h}|}$. Then, by Corollary 3.4, ${\boldsymbol{p}}^{\prime}$ ${\text{MT}(\vv\boldsymbol{\pi}^{\prime})}$-forces $\varphi[{\underline{G}}]$, but ${\boldsymbol{q}}^{\prime}$ ${\text{MT}(\vv\text{\leavevmode{}}^{\prime})}$-forces $\psi[{\underline{G}}]$, over ${{\mathfrak{M}}}$. However the sets $\text{MT}(\vv\boldsymbol{\pi}^{\prime})$, $\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ belong to the same model ${{\mathfrak{M}}}$, where they are order-isomorphic via the isomorphism induced by $\boldsymbol{h}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}|\boldsymbol{h}|$. Therefore, and since ${\boldsymbol{q}}=\boldsymbol{h}{\boldsymbol{p}}$ and $\psi=\boldsymbol{h}\varphi$, it cannot happen that both ${\boldsymbol{p}}\,$ ${\text{MT}(\vv\boldsymbol{\pi}^{\prime})}$-forces $\varphi[{\underline{G}}]$ and ${\boldsymbol{q}}\,$ ${\text{MT}(\vv\text{\leavevmode{}}^{\prime})}$-forces $\neg\>\psi[{\underline{G}}]$. But this contradicts to the above.

(ii) Consider first the $\mathscr{L}\varPi^{1}_{2}$ case. Assume that $\varphi(x)$ is a $\mathscr{L}\varSigma^{1}_{1}$ formula, $\psi(x):=\boldsymbol{h}\varphi(x)$, ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}{\forall\,}x\,\varphi(x)$, but to the contrary ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}{\forall\,}x\,\psi(x)$ fails. Thus there is a multisequence $\vv\text{\leavevmode{}}^{\prime}\in\vv\text{sMF}$ and a multitree ${\boldsymbol{q}}^{\prime}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ such that $\vv\text{\leavevmode{}}\subset\vv\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{q}}^{\prime}\leqslant{\boldsymbol{q}}$, and ${\boldsymbol{q}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}{\exists\,}x\,\psi^{-}(x)$. By definition there is a small real name $\mathbf{d}$ such that ${\boldsymbol{q}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\psi^{-}(\mathbf{d})$. The multisequence $\vv\boldsymbol{\pi}^{\prime}=\boldsymbol{h}^{-1}\vv\text{\leavevmode{}}^{\prime}$ then satisfies $\vv\boldsymbol{\pi}\subset\vv\text{\leavevmode{}}$, the multitree ${\boldsymbol{p}}^{\prime}=\boldsymbol{h}^{-1}{\boldsymbol{q}}^{\prime}$ belongs to $\text{MT}(\vv\boldsymbol{\pi}^{\prime})$ and ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, $\mathbf{c}=\boldsymbol{h}^{-1}{\mathbf{d}}$ is a small real name, and we have ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt wforc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\varphi^{-}(\mathbf{c})$ by (i). Then by Lemma 3.2 there is a longer multisequence $\vv\boldsymbol{\sigma}\in\vv\text{sMF}$ satisfying $\vv\boldsymbol{\pi}^{\prime}\subset\vv\boldsymbol{\sigma}$ and ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\sigma}}\hskip 1.72218pt}\varphi^{-}(\mathbf{c})$, that is, we have ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\sigma}}\hskip 1.72218pt}{\exists\,}x\,\varphi^{-}(x)$. But by definition ($3^{\circ}$ in Section 2) this contradicts to the assumption ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}{\forall\,}x\,\varphi(x)$.

To carry out the step $\mathscr{L}\varPi^{1}_{n}\to\mathscr{L}\varSigma^{1}_{n+1}$, $n\geq 2$, let $\varphi(x)$ be a formula in $\mathscr{L}\varPi^{1}_{n}$, $\psi(x):=\boldsymbol{h}\varphi(x)$, and $\boldsymbol{h}\in\text{\rm PERM}_{n-2}$. Assume that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}{\exists\,}x\,\varphi(x)$. By definition (see $2^{\circ}$ in Section 2), there is a small real name $\mathbf{c}$ such that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi(\mathbf{c})$. Then we have ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\psi(\mathbf{d})$ by inductive assumption, where $\mathbf{d}=\boldsymbol{h}\mathbf{c}$ is a small real name itself. Thus ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}{\exists\,}x\,\psi(x)$.

To carry out the step $\mathscr{L}\varSigma^{1}_{n}\to\mathscr{L}\varPi^{1}_{n}$, $n\geq 3$, let $\varphi$ be a formula in $\mathscr{L}\varPi^{1}_{n}$, and $\boldsymbol{h}\in\text{\rm PERM}_{n-2}$. Let ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, in particular $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, but, to the contrary, ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\psi$ fails, where ${\boldsymbol{q}}=\boldsymbol{h}{\boldsymbol{p}}$, $\vv\text{\leavevmode{}}=\boldsymbol{h}\vv\boldsymbol{\pi}$, and $\psi$ is $\boldsymbol{h}\varphi$, as above. Then in our assumptions, ${\vv\text{\leavevmode{}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}={\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-2}$, hence $\vv\text{\leavevmode{}}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ as well. Therefore by definition ($3^{\circ}$ in Section 2) there is a multisequence $\vv\text{\leavevmode{}}^{\prime}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ and ${\boldsymbol{q}}^{\prime}\in\text{MT}(\vv\text{\leavevmode{}}^{\prime})$ such that $\vv\text{\leavevmode{}}\subseteq\vv\text{\leavevmode{}}^{\prime}$, ${\boldsymbol{q}}^{\prime}\leqslant{\boldsymbol{q}}$, and ${\boldsymbol{q}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}^{\prime}}\hskip 1.72218pt}\psi^{-}$.

Now let ${\boldsymbol{p}}^{\prime}=\boldsymbol{h}^{-1}{\boldsymbol{q}}^{\prime}$ and $\vv\boldsymbol{\pi}^{\prime}=\boldsymbol{h}^{-1}\vv\text{\leavevmode{}}^{\prime}$, so that ${\boldsymbol{p}}^{\prime}\leqslant{\boldsymbol{p}}$, $\vv\boldsymbol{\pi}\subseteq\vv\boldsymbol{\pi}^{\prime}$, and, that is most important, $\vv\boldsymbol{\pi}^{\prime}$ belongs to $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ since so does $\vv\text{\leavevmode{}}^{\prime}$ and $\boldsymbol{h}^{-1}\in\text{\rm PERM}_{n-2}$. Moreover we have ${\boldsymbol{p}}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}^{\prime}}\hskip 1.72218pt}\varphi^{-}$ by inductive assumption. We conclude that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$ fails, which is a contradiction. ∎

6 Forcing with subsequences of the key multisequence

We argue in ${\mathbf{L}}$. The key multisequence $\vv{\nmi\Pi}=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\omega_{1}}\in\vv\text{sMF}_{\omega_{1}}$, satisfying (i), (ii), (iii) of Theorem 4.1, was fixed by 4.4, and $\boldsymbol{\mathbb{P}}=\text{MT}(\vv{\nmi\Pi})$ is our forcing notion. If $\gamma<\omega_{1}$ then the subsequence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ belongs to $\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$, ${\forall\,}m$.

Definition 6.1**.**

We write ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\alpha}\hskip 1.72218pt}\varphi$ instead of ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\alpha}\hskip 1.72218pt}\varphi$, for the sake of brevity. Let ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi$ mean: ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\alpha}\hskip 1.72218pt}\varphi$ for some $\alpha<\omega_{1}$. ∎

Lemma 6.2** (in ${\mathbf{L}}$).**

Assume that ${\boldsymbol{p}}\in\boldsymbol{\mathbb{P}}$, $\alpha<\omega_{1}$, and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\alpha}\hskip 1.72218pt}\varphi$. Then$:$

1. (i)

if $\alpha\leq\beta<\omega_{1}$, ${\boldsymbol{q}}\in\boldsymbol{\mathbb{P}}_{<\beta}=\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\beta)$, and ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, then ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\beta}\hskip 1.72218pt}\varphi\;;$ 2. (ii)

if ${\boldsymbol{q}}\in\boldsymbol{\mathbb{P}}$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, then ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\beta}\hskip 1.72218pt}\varphi$ for some $\beta\,;$ $\alpha\leq\beta<\omega_{1}\;;$ 3. (iii)

if ${\boldsymbol{q}}\in\boldsymbol{\mathbb{P}}$ and ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi^{-}$ then ${\boldsymbol{p}},{\boldsymbol{q}}$ are somewhere almost disjoint*;*** 4. (iv)

therefore, 1st, if ${\boldsymbol{p}},{\boldsymbol{q}}\in\boldsymbol{\mathbb{P}}$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi$ then ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi$, and 2nd, ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi$, ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi^{-}$ cannot hold together.

Proof**.**

To prove (i) apply Lemma 2.3. To prove (ii) let $\beta$ satisfy $\alpha<\beta<\omega_{1}$ and ${\boldsymbol{q}}\in\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\beta)$, and apply (i). Finally to prove (iii) note that ${\boldsymbol{p}},{\boldsymbol{q}}$ have to be incompatible in $\boldsymbol{\mathbb{P}}$, as otherwise (i) leads to contradiction, but the incompatibility in $\boldsymbol{\mathbb{P}}$ implies being somewhere a. d. by Corollary 4.1. ∎

Now we are going to prove that the auxiliary relation $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$ essentially coincides with the usual ${\boldsymbol{\mathbb{P}}}$-forcing relation over ${\mathbf{L}}$.

Lemma 6.3**.**

If $n<\omega$, $\varphi$ is a closed formula as in 2.2, and ${\boldsymbol{p}}\in\boldsymbol{\mathbb{P}}$, then ${\boldsymbol{p}}$ ${\boldsymbol{\mathbb{P}}}$-forces $\varphi[{\underline{G}}]$ over ${\mathbf{L}}$ in the usual sense if and only if ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi$.

Proof**.**

Let $\mathrel{{\|}{-}}$ denote the usual ${\boldsymbol{\mathbb{P}}}$-forcing relation over ${\mathbf{L}}$.

Part 1: $\varphi$ is a formula in $\mathscr{L}(\varSigma{+}\varPi)^{1}_{1}$. If ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi$ then ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi$ for some $\gamma<\omega_{1}$, and then ${\boldsymbol{p}}\mathrel{{\|}{-}}\varphi[{\underline{G}}]$ by Lemma 3.1 with $\vv\text{\leavevmode{}}=\vv{\nmi\Pi}$ and ${{\mathfrak{N}}}={\mathbf{L}}$.

Suppose now that ${\boldsymbol{p}}\mathrel{{\|}{-}}\varphi[{\underline{G}}]$. There is an ordinal $\gamma_{0}<\omega_{1}$ such that ${\boldsymbol{p}}\in\boldsymbol{\mathbb{P}}_{\gamma_{0}}=\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0})$ and $\varphi$ belongs to ${{\mathfrak{M}}}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0})$. The set $U$ of all multitrees $\vv\boldsymbol{\pi}\in\vv\text{sMF}$ such that $\gamma_{0}<\mathop{\tt dom}\vv\boldsymbol{\pi}$ and there is an ordinal $\vartheta$, $\gamma_{0}<\vartheta<\mathop{\tt dom}\vv\boldsymbol{\pi}$, such that $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta\subset_{{{\mathfrak{M}}}(\vv\boldsymbol{\pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\vartheta)}\vv\boldsymbol{\pi}$, is dense in $\vv\boldsymbol{\pi}$ by Corollary 1.2(ii), and is ${\mathbf{\Delta}}^{\mathrm{HC}}_{1}$. Therefore by Corollary 4.5 there is an ordinal $\gamma<\omega_{1}$ such that $\vv\boldsymbol{\pi}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\in U$.

Let this be witnessed by an ordinal $\vartheta$, so that $\gamma_{0}<\vartheta<\gamma=\mathop{\tt dom}\vv\boldsymbol{\pi}$ and $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta\subset_{{{\mathfrak{M}}}(\vv\boldsymbol{\pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\vartheta)}\vv\boldsymbol{\pi}$. We claim that ${\boldsymbol{p}}$ ${\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)}$-forces $\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$ in the usual sense — then by definition ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$, and we are done.

To prove the claim, suppose otherwise. Then there is a multitree ${\boldsymbol{q}}\in\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, which ${\text{MT}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)}$-forces $\neg\>\varphi[{\underline{G}}]$ over ${{\mathfrak{M}}}(\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\vartheta)$. Then by definition we have ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\neg\,\varphi$, hence ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\neg\,\varphi$, which implies ${\boldsymbol{q}}\mathrel{{\|}{-}}\neg\>\varphi[{\underline{G}}]$ (see above), with a contradiction to ${\boldsymbol{p}}\mathrel{{\|}{-}}\varphi[{\underline{G}}]$.

Part 2: the step $\mathscr{L}\varPi^{1}_{n}\to\mathscr{L}\varSigma^{1}_{n+1}$ ($n\geq 1$). Consider a $\mathscr{L}\varPi^{1}_{n}$ formula $\varphi(x)$. Assume ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}{\exists\,}x\,\varphi(x)$. By definition there is a small real name $\mathbf{c}$ such that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi(\mathbf{c})$. By inductive hypothesis, ${\boldsymbol{p}}\mathrel{{\|}{-}}\varphi(c)[{\underline{G}}]$, that is, ${\boldsymbol{p}}\mathrel{{\|}{-}}{\exists\,}x\,\varphi(x)[{\underline{G}}]$. Conversely, assume that ${\boldsymbol{p}}\mathrel{{\|}{-}}{\exists\,}x\,\varphi(x)[{\underline{G}}]$. As $\boldsymbol{\mathbb{P}}$ is CCC, there is a small real name $\mathbf{c}$ (in ${\mathbf{L}}$) such that ${\boldsymbol{p}}\mathrel{{\|}{-}}\varphi(\mathbf{c})[{\underline{G}}]$. We have ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi(\mathbf{c})$ by the inductive hypothesis, hence ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}{\exists\,}x\,\varphi(x)$.

Part 3: the step $\mathscr{L}\varSigma^{1}_{n}\to\mathscr{L}\varPi^{1}_{n}$ ($n\geq 2$). Consider a closed $\mathscr{L}\varSigma^{1}_{n}$ formula $\varphi$. Assume that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi^{-}$. By Lemma 6.2(iv), there is no multitree ${\boldsymbol{q}}\in\boldsymbol{\mathbb{P}}$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, with ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi$. This implies ${\boldsymbol{p}}\mathrel{{\|}{-}}\varphi^{-}$ by the inductive hypothesis.

Conversely, suppose that ${\boldsymbol{p}}\mathrel{{\|}{-}}\varphi^{-}$. There is an ordinal $\gamma_{0}<\omega_{1}$ such that ${\boldsymbol{p}}\in\boldsymbol{\mathbb{P}}_{\gamma_{0}}=\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0})$ and $\varphi$ belongs to ${{\mathfrak{M}}}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0})$. Consider the set $U$ of all multisequences of the form ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptn-2}$, where $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$, $\mathop{\tt dom}\vv\boldsymbol{\pi}>\gamma_{0}$, and there is a multitree ${\boldsymbol{q}}\in\text{MT}(\vv\boldsymbol{\pi})$ satisfying ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$ (${\boldsymbol{q}}$ is stronger) and ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi$. It follows from Lemma 1.2 and Lemma 2.4 that $U$ belongs to ${\mathbf{\Sigma}}^{\mathrm{HC}}_{n-1}$ (with $\varphi$ and ${\boldsymbol{p}}_{0}$ as parameters). Therefore by 4.4(ii) there is an ordinal $\gamma<\omega_{1}$ such that the subsequence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$  ${(n-2)}$-decides $U$.

Case 1: ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptn-2}\in U$. Let this be witnessed by a multisequence $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ and a multitree ${\boldsymbol{q}}\in\text{MT}(\vv\boldsymbol{\pi})$, so that in particular ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptn-2}={\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptn-2}$ and $\mathop{\tt dom}\vv\boldsymbol{\pi}=\gamma>\gamma_{0}$. Then by definition (Definition 1.1) we also have ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}={(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}$, so that overall $\vv\boldsymbol{\pi}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$. Thus ${\boldsymbol{q}}\in\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi$, that is, ${\boldsymbol{q}}\mathrel{{\|}{-}}\varphi[{\underline{G}}]$ by the inductive hypothesis, contrary to the choice of ${\boldsymbol{p}}$. Therefore Case 1 cannot happen, and we have:

Case 2: negative decision, no multisequence in $U$ extends ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptn-2}$. We can assume that $\gamma>\gamma_{0}$. (Otherwise replace $\gamma$ by $\gamma_{0}+1$.) We claim that ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\gamma}\hskip 1.72218pt}\varphi^{-}$. Indeed otherwise by $3^{\circ}$ there is a multisequence $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptn-2}]$ and a multitree ${\boldsymbol{q}}\in\text{MT}(\vv\boldsymbol{\pi})$, such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subseteq\vv\boldsymbol{\pi}$, ${\boldsymbol{q}}\leqslant{\boldsymbol{p}}$, and ${\boldsymbol{q}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\text{\leavevmode{}}}\hskip 1.72218pt}\varphi$. But then $\vv\boldsymbol{\pi}$ and ${\boldsymbol{q}}$ witness that $\vv\boldsymbol{\sigma}={\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptn-2}$ belongs to $U$. On the other hand, $\vv\boldsymbol{\sigma}$ obviously extends ${\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptn-2}$, since $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subseteq\vv\boldsymbol{\pi}$, contrary to the Case 2 assumption. Thus indeed ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi^{-}$, as required. ∎

The next lemma provides a useful strengthening.

Lemma 6.4**.**

If $\Phi$ is a $\varDelta^{\mathrm{HC}}_{1}$ collection of closed $\varPi^{1}_{n+2}$ formulas, ${\boldsymbol{p}}_{0}\in\boldsymbol{\mathbb{P}}$, and ${\boldsymbol{p}}_{0}$ ${\boldsymbol{\mathbb{P}}}$-forces $\varphi[{\underline{G}}]$ over ${\mathbf{L}}$ for each $\varphi\in\Phi$, then there is an ordinal $\gamma<\omega_{1}$ such that if $\varphi\in\Phi$ then ${\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi$. (Same $\gamma$ for all $\varphi$.)

Proof**.**

Let $U$ consist of all multisequences of the form ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, where $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$, and there is a formula $\varphi\in\Phi$ and ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$ such that ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}$ and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi^{-}$. It follows from Lemma 1.2 and 2.4 that $U$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m+1}$ set, so by 4.4(ii) there is an ordinal $\gamma<\omega_{1}$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$  ${m}$-decides $U$.

Case 1: ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in U$, that is, the multisequence $\vv\boldsymbol{\pi}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ satisfies the condition that there exist $\varphi\in\Phi$ and a multitree ${\boldsymbol{p}}\in\vv\boldsymbol{\pi}$ such that ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}$ and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi^{-}$, and hence ${\boldsymbol{p}}\,$ ${\boldsymbol{\mathbb{P}}}$-forces $\varphi^{-}[{\underline{G}}]$ over ${\mathbf{L}}$ by Lemma 6.3, contrary to the choice of ${\boldsymbol{p}}_{0}$. Therefore Case 1 cannot happen, and we have

Case 2: no multisequence in $U$ extends ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$. We can assume that $\gamma>\gamma_{0}$. (Otherwise replace $\gamma$ by $\gamma_{0}+1$.) We claim that $\gamma$ is as required. Indeed otherwise ${\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi$ fails for a formula $\varphi\in\Phi$, thus ($3^{\circ}$ in Section 2), there is a multisequence $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ and ${\boldsymbol{p}}\in\text{MT}(\vv\boldsymbol{\pi})$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subseteq\vv\boldsymbol{\pi}$, ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}$, and ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv\boldsymbol{\pi}}\hskip 1.72218pt}\varphi^{-}$. It follows that ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in U$. In addition, ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ extends ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ by construction. But this contradicts to the Case 2 assumption. ∎

VI The model

In this conclusive section we gather the results obtained above towards the proof of Theorem 1.1. We begin with the analysis of definability of key generic reals in ${\boldsymbol{\mathbb{P}}}$-generic extensions of ${\mathbf{L}}$, which will lead to (I) and (II) of Theorem 1.1. Then we proceed to (III) (elementary equivalence) and (IV) (the non-wellorderability).

1 Key generic extension and subextensions

Recall that the key multisequence $\vv{\nmi\Pi}=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\omega_{1}}$ of small multiforcings $\nmi\Pi_{\alpha}$ is defined in ${\mathbf{L}}$ by 4.4, $\nmi\Pi=\bigcup^{\text{\tt cw}}_{\alpha<\omega_{1}}$ is a multiforcing, $|\nmi\Pi|=\omega_{1}\times\omega$ in ${\mathbf{L}}$, and $\boldsymbol{\mathbb{P}}=\text{MT}(\vv{\nmi\Pi})=\text{MT}(\nmi\Pi)\in{\mathbf{L}}$ is our key forcing notion, equal to the finite-support product $\prod_{\xi<\omega_{1},k<\omega}\nmi\Pi(\xi,k)$ of perfect-tree forcings $\nmi\Pi(\xi,k)$ in ${\mathbf{L}}$. See Section 5, where some properties of $\boldsymbol{\mathbb{P}}$ are established, including CCC and definability of the factors $\nmi\Pi(\xi,k)$.

From now on, we’ll typically argue in ${\mathbf{L}}$ and in ${\boldsymbol{\mathbb{P}}}$-generic extensions of ${\mathbf{L}}$, so by Lemma 5.6 it will always be true that $\omega_{1}^{\mathbf{L}}=\omega_{1}$. This allows us to still think that $|\nmi\Pi|=\omega_{1}\times\omega$ (rather than $\omega_{1}^{\mathbf{L}}\times\omega$).

Recall that $\nmi\Pi\in{\mathbf{L}}$ and $\boldsymbol{\mathbb{P}}=\text{MT}(\nmi\Pi)$ is a forcing notion in ${\mathbf{L}}$.

Definition 1.1**.**

Let a set $G\subseteq\boldsymbol{\mathbb{P}}$ be generic over the constructible set universe ${\mathbf{L}}$. If $\langle\xi,k\rangle\in\omega_{1}\times\omega$ then following Remark 4.3, we

• $-$

define $G(\xi,k)=\{\hskip 0.43057pt{T^{{\boldsymbol{p}}}_{\xi k}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt{\boldsymbol{p}}\in G\land\langle\xi,k\rangle\in|{\boldsymbol{p}}|\hskip 0.43057pt\}\subseteq\nmi\Pi(\xi,k)$;

• $-$

let $x_{\xi k}=x_{\xi k}[G]\in 2^{\omega}$ be the only real in $\bigcap_{T\in G(\xi,k)}[T]$.

Thus $\boldsymbol{\mathbb{P}}$ adds an array $X=\langle x_{\xi k}\rangle_{\langle\xi,k\rangle\in\omega_{1}\times\omega}$ of reals, where each real $x_{\xi k}=x_{\xi k}[G]\in\in 2^{\omega}\cap{\mathbf{L}}[G]$ is a ${\nmi\Pi(\xi,k)}$-generic real over ${\mathbf{L}}$, and ${\mathbf{L}}[G]={\mathbf{L}}[X]$. ∎

Let $G\subseteq\boldsymbol{\mathbb{P}}$ be a set (filter) ${\boldsymbol{\mathbb{P}}}$-generic over ${\mathbf{L}}$. If $m<\omega$ then following the notation in Section 2 we define

[TABLE]

so that the set ${G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ is ${{\boldsymbol{\mathbb{P}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}}$-generic

over ${\mathbf{L}}$, where accordingly

[TABLE]

Each subextension ${\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]\subseteq{\mathbf{L}}[G]$ coincides with ${\mathbf{L}}[\langle x_{\xi k}[G]\rangle_{\xi<\omega_{1}\land k<m}]$. Our goal will be to demonstrate that the model ${\mathbf{L}}[X]={\mathbf{L}}[G]$, along with the system of submodels ${\mathbf{L}}[\langle x_{\xi k}[G]\rangle_{\xi<\omega_{1}\land k<m}]$, proves Theorem 1.1.

2 Definability of generic reals

Recall that the factors $\nmi\Pi(\xi,k)$ of the forcing notion $\nmi\Pi$ are defined by $\nmi\Pi(\xi,k)=\bigcup_{\alpha(\xi,k)\leq\alpha<\omega_{1}}\nmi\Pi_{\alpha}(\xi,k)$, where $\alpha(\xi,k)<\omega_{1}$, the sets $\nmi\Pi_{\alpha}(\xi,k)$ are countable sets of perfect trees, whose definability in ${\mathbf{L}}$ is determined by Corollary 5.3.

Theorem 2.1**.**

Assume that a set $G\subseteq\boldsymbol{\mathbb{P}}$ is ${\boldsymbol{\mathbb{P}}}$-generic over ${\mathbf{L}}$, $\xi<\omega_{1}$, $k<\omega$, and $x\in{\mathbf{L}}[G]\cap\omega^{\omega}.$ The following are equivalent$:$

1. (1)

$x=x_{\xi k}[G]\;;$** 2. (2)

$x$* is ${\nmi\Pi(\xi,k)}$-generic over ${\mathbf{L}}\;;$* 3. (3)

$x\in\bigcap_{\alpha(\xi,k)\leq\alpha<\omega_{1}}\bigcup_{T\in\nmi\Pi_{\alpha}(\xi,k)}[T]$.

Proof**.**

$\ref{xik1}\mathbin{\,\Longrightarrow\,}\ref{xik2}$ is a routine (see Remark 4.3). To check $\ref{xik2}\mathbin{\,\Longrightarrow\,}\ref{xik3}$ recall that each set $\nmi\Pi_{\alpha}(\xi,k)$ is pre-dense in $\nmi\Pi(\xi,k)$ by Lemma 1.2(v). It remains to establish $\ref{xik3}\mathbin{\,\Longrightarrow\,}\ref{xik1}$. Suppose towards the contrary that a real $x\in{\mathbf{L}}[G]\cap 2^{\omega}$ satisfies (3) but $x\neq x_{\xi k}[G]$. By Theorem 6.2(i) there is a true ${\nmi\Pi}$-real name $\mathbf{c}=\langle C_{ni}\rangle_{n,i<\omega}$, non-principal over $\nmi\Pi$ at $\xi,k$ and such that $x=\mathbf{c}[G]$. Being non-principal means that the next set is open dense in $\boldsymbol{\mathbb{P}}=\text{MT}(\nmi\Pi)$:

[TABLE]

And as $\boldsymbol{\mathbb{P}}=\text{MT}(\nmi\Pi)$ is a CCC forcing by Lemma 5.6, we can assume that the name $\mathbf{c}$ is small, that is, each set $C_{ni}\subseteq\boldsymbol{\mathbb{P}}$ is countable. Then there is an ordinal $\gamma_{0}<\omega_{1}$ such that $C_{ni}\subseteq\boldsymbol{\mathbb{P}}_{<\gamma_{0}}$ for all $n,i$. Then $\mathbf{c}$ is a true ${\nmi\Pi_{<\gamma_{0}}}$-real name. Moreover we can assume by Corollary 5.7 that ${\boldsymbol{D}}^{\nmi\Pi}_{\xi k}(\mathbf{c})\cap\boldsymbol{\mathbb{P}}_{<\gamma_{0}}$ is pre-dense in $\boldsymbol{\mathbb{P}}$.

Now consider the set $W$ of all multisequences $\vv\boldsymbol{\pi}=\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\mathop{\tt dom}(\vv\boldsymbol{\pi})}\in\vv\text{sMF}$ such that $\mathop{\tt dom}(\vv\boldsymbol{\pi})>\gamma_{0}$ and

• $-$

either (I) $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}\not\subset\vv\boldsymbol{\pi}$;

• $-$

or (II) $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}\subset\vv\boldsymbol{\pi}$ and $\mathbf{c}$ is not non-principal over $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}\vv\boldsymbol{\pi}$ at $\xi,k$;

• $-$

or (III) $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}\subset\vv\boldsymbol{\pi}$, $\mathop{\tt dom}(\vv\boldsymbol{\pi})=\delta+1$ is a successor, and $\bigcup^{\text{\tt cw}}_{\alpha<\delta}\boldsymbol{\pi}_{\alpha}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\boldsymbol{\pi}_{\delta}$.

We assert that $W$ is dense in $\vv\text{sMF}$: any multisequence $\vv\boldsymbol{\pi}\in\vv\text{sMF}$ can be extended to some $\vv\text{\leavevmode{}}\in W$. Indeed first extend $\vv\boldsymbol{\pi}$ by Corollary 1.2 so that is has a length $\mathop{\tt dom}(\vv\boldsymbol{\pi})=\delta>\gamma_{0}$. If now $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}\not\subset\vv\boldsymbol{\pi}$ then immediately $\vv\boldsymbol{\pi}\in W$ via (I), so we assume that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}\subset\vv\boldsymbol{\pi}$. We can also assume that $\mathbf{c}$ is non-principal over $\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}\vv\boldsymbol{\pi}$ at $\xi,k$ by similar reasons related to (II). The multisequence $\vv\boldsymbol{\pi}$ can be extended, by Corollary 1.2, by an extra term $\boldsymbol{\pi}_{\delta}$, so that the extended multisequence $\vv\boldsymbol{\pi}_{+}$ satisfies $\vv\boldsymbol{\pi}\subset_{\{\hskip 0.3014pt\mathbf{c}\hskip 0.3014pt\}}\vv\boldsymbol{\pi}_{+}$, that is, $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{\{\hskip 0.3014pt\mathbf{c}\hskip 0.3014pt\}}\boldsymbol{\pi}_{\delta}$. By definition (Definition 7.1) and the nonprincipality of $\mathbf{c}$, we get $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\boldsymbol{\pi}_{\delta}$. It follows that $\vv\boldsymbol{\pi}_{+}\in W$ via (III).

Since $W$ is ${\mathbf{\Sigma}}^{\mathrm{HC}}_{1}$, by Definition 4.4(ii) there is an ordinal $\gamma<\omega_{1}$ such that the multisequence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ ${0}$-decides $W$. However the negative decision is impossible by the density (see the proof of Lemma 5.2). We conclude that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\in W$; hence, $\gamma>\gamma_{0}$. Option (I) for $\vv\boldsymbol{\pi}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ clearly fails, and (II) fails either because the set ${\boldsymbol{D}}^{\nmi\Pi}_{\xi k}(\mathbf{c})\cap\boldsymbol{\mathbb{P}}_{<\gamma_{0}}$ is pre-dense in $\boldsymbol{\mathbb{P}}$ and $\gamma>\gamma_{0}$. Therefore $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ belongs to $W$ via (III), that is, $\gamma=\delta+1$ and $\nmi\Pi_{<\delta}=\bigcup^{\text{\tt cw}}_{\alpha<\delta}\nmi\Pi_{\alpha}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\nmi\Pi_{\delta}$. Then $\nmi\Pi_{<\delta}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\nmi\Pi_{\geq\delta}=\bigcup^{\text{\tt cw}}_{\delta\leq\alpha<\omega_{1}}\nmi\Pi_{\delta}$ by Lemma 4.3(iii).

Now we make use of Theorem 6.2(ii) with $\boldsymbol{\pi}=\nmi\Pi_{<\delta}$ and $\text{\leavevmode{}}=\nmi\Pi_{\geq\delta}$; note that $\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}}=\nmi\Pi$. It follows that $x=\mathbf{c}[G]\notin\bigcup_{Q\in\nmi\Pi_{\geq\delta}(\xi,k)}[Q]$, which clearly contradicts to the assumption (3). ∎

Corollary 2.2**.**

Assume that $k<\omega$ and $G\subseteq\boldsymbol{\mathbb{P}}$ is ${\boldsymbol{\mathbb{P}}}$-generic over ${\mathbf{L}}$. Then

[TABLE]

is a set of definability class $\varPi^{\mathrm{HC}}_{k+2}$ in ${\mathbf{L}}[G]$ and in any transitive model $M\models\text{ZFC}$ satisfying ${\mathbf{L}}\subseteq M\subseteq{\mathbf{L}}[G]$ and $\{\hskip 0.43057pt{x_{\xi k}[G]\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\xi<\omega_{1}\hskip 0.43057pt\}\subseteq M$.

Proof**.**

By the theorem, it is true in ${\mathbf{L}}[G]$ that $\langle\xi,x\rangle\in\mathbb{W}_{k}$ iff

[TABLE]

which can be re-written as

[TABLE]

Here the equality $\mu=\alpha(\xi,k)$ (with a fixed $k$) is $\varDelta^{\mathrm{HC}}_{k+2}$ by Corollary 5.3, and so is the equality $X=\nmi\Pi_{\alpha}(\xi,k)$ by Corollary 5.3. It follows that the whole relation is $\varPi^{\mathrm{HC}}_{k+2}$, since the quantifier ${\exists\,}T\in X$ is bounded. ∎

The next corollary is the first cornerstone in the proof of Theorem 1.1.

Corollary 2.3** (= (I), (II) of Theorem 1.1).**

Assume that $m<\omega$ and a set $G\subseteq\boldsymbol{\mathbb{P}}$ is ${\boldsymbol{\mathbb{P}}}$-generic over ${\mathbf{L}}$. Then $\omega^{\omega}\cap{\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ is a $\varSigma^{1}_{m+3}$ set in ${\mathbf{L}}[G]$, and it holds in ${\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ that there is a $\varDelta^{1}_{m+3}$ wellordering of $\omega^{\omega}$ of length $\omega_{1}$.

Proof**.**

If $\gamma<\omega_{1}$ then let $X_{\gamma n}=\langle x_{\xi k}[G]\rangle_{\xi<\gamma\land k<n}$; thus $X=X_{\gamma n}$ is a $\varPi^{\mathrm{HC}}_{n+1}$ relation in ${\mathbf{L}}[G]$ (with $\gamma,n,X$ as arguments) by Corollary 2.2. It follows that

[TABLE]

is a set in $\varSigma^{\mathrm{HC}}_{n+2}$, hence, a $\varSigma^{1}_{m+3}$ set in ${\mathbf{L}}[G]$. To define a required wellordering, if $x\in\omega^{\omega}\cap{\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ then let $\gamma(x)$ be the least $\gamma<\omega_{1}$ such that $x\in{\mathbf{L}}[X_{\gamma n}]$, and let $\nu(x)<\omega_{1}$ be the index of $x$ in the canonical wellordering of $\omega^{\omega}$ in ${\mathbf{L}}[X_{\gamma n}]$. Now we wellorder $\omega^{\omega}\cap{\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ according to the lexicographical ordering of triples $\langle\mathop{\tt max}\{\hskip 0.43057pt\gamma(x),\nu(x)\hskip 0.43057pt\},\gamma(x),\nu(x)\rangle$. ∎

3 Elementary equivalence

Here we prove the following elementary equivalence theorem for key generic extensions. Compare to (III) of Theorem 1.1.

Theorem 3.1**.**

Assume that $m<\omega$ and a set $G\subseteq\boldsymbol{\mathbb{P}}$ is ${\boldsymbol{\mathbb{P}}}$-generic over ${\mathbf{L}}$. Then ${\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ is an elementary submodel of ${\mathbf{L}}[G]$ w. r. t. all $\varSigma^{1}_{m+2}$ formulas.

Proof**.**

Suppose that this is not the case. Then there is a $\varPi^{1}_{m+1}$ formula $\varphi(r,x)$ with $r\in\omega^{\omega}\cap{\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ as the only parameter, and a real $x_{0}\in\omega^{\omega}\cap{\mathbf{L}}[G]$ such that $\varphi(r,x_{0})$ is true in ${\mathbf{L}}[G]$ but there is no $x\in\omega^{\omega}\cap{\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ such that $\varphi(r,x)$ is true in ${\mathbf{L}}[G]$. By a version of Proposition 6.1(ii), we have $r=\mathbf{c}_{0}[G]$, where $\mathbf{c}_{0}$ is a small true ${({\boldsymbol{\mathbb{P}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm})}$-real name. (See Section 1 on notation.) And there is a small true ${\boldsymbol{\mathbb{P}}}$-real name $\mathbf{c}$ such that $x_{0}=\mathbf{c}[{\underline{G}}]$.

By Lemma 6.3, there is a multitree ${\boldsymbol{p}}_{0}\in G$ such that

1. (1)

${\boldsymbol{p}}_{0}$  ${\boldsymbol{\mathbb{P}}}$-forces $\varphi(\mathbf{c}_{0}[{\underline{G}}],\mathbf{c}[{\underline{G}}])\,\land\,\neg\>{\exists\,}x\in{\mathbf{L}}[{{\underline{G}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]\,\varphi(\mathbf{c}_{0}[{\underline{G}}],x)$ over ${\mathbf{L}}$; 2. (2)

${\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}\varphi(\mathbf{c}_{0},\mathbf{c})$, that is, ${\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma_{0}}\hskip 1.72218pt}\varphi(\mathbf{c}_{0},\mathbf{c})$, where $\gamma_{0}<\omega_{1}$ — and we can assume that ${\boldsymbol{p}}_{0}\in\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0})$ as well.

As $\mathbf{c},\mathbf{c}_{0}$ are small names, there is an ordinal $\delta<\omega_{1}$ satisfying

1. (3)

$|\mathbf{c}_{0}|\subseteq\delta\times m$, $|\mathbf{c}|\subseteq\delta\times\omega$, and $|{\boldsymbol{p}}_{0}|\subseteq\delta\times\omega$,

and we can enlarge $\gamma_{0}$, if necessary, using the equality $|\vv{\nmi\Pi}|=\omega_{1}\times\omega$ of Lemma 5.2, to make sure that

1. (4)

$\delta\times\omega\subseteq|\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}|$, that is, if $\eta<\delta$ and $k<\omega$ then $\langle\eta,k\rangle\in|\nmi\Pi_{\alpha^{\prime}}|$ for some $\alpha^{\prime}=\alpha^{\prime}(\eta,k)<\gamma_{0}$.

We are starting from here towards a contradiction.

Let $U$ consist of all multisequences of the form ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, where

1. (A)

$\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$, $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}\subset\vv\boldsymbol{\pi}$, and hence ${\boldsymbol{p}}_{0}\in\text{MT}(\vv\boldsymbol{\pi})$ by (2);

and there is an ordinal $\zeta<\omega_{1}$ and a transformation $\boldsymbol{h}\in\text{\rm PERM}_{m-1}$ such that

1. (B)

$\boldsymbol{h}=\boldsymbol{h}^{-1}$, $|\boldsymbol{h}|=D\cup R$, and $\boldsymbol{h}$ maps $D$ onto $R$ and $R$ onto $D$, where $D=\delta\times[m,\omega)$, $R=\{\hskip 0.43057pt{\langle\xi,m-1\rangle\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\nu_{0}\leq\xi<\nu_{1}\hskip 0.43057pt\}$, and $\delta<\nu_{0}<\nu_{1}<\omega_{1}$; 2. (C)

$\gamma_{0}\leq\zeta<\mathop{\tt dom}\vv\boldsymbol{\pi}$ and $(\boldsymbol{h}\vv\boldsymbol{\pi}){{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\zeta}=\vv\boldsymbol{\pi}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\zeta}$, or equivalently $\boldsymbol{h}(\vv\boldsymbol{\pi}(\alpha))=\vv\boldsymbol{\pi}(\alpha)$ whenever $\zeta\leq\alpha<\mathop{\tt dom}\vv\boldsymbol{\pi}$.

It follows from Lemma 1.2 that $U$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m+1}$ set (with $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0}$, $\delta$ as parameters). Therefore by 4.4(ii) there is an ordinal $\gamma<\omega_{1}$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$  ${m}$-decides $U$.

Case 1: ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in U$. Basically this means that there is a transformation $\boldsymbol{h}\in\text{\rm PERM}_{m-1}$ such that (A), (B), (C) hold for $\boldsymbol{h}$ and $\vv\boldsymbol{\pi}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$, via ordinals $\delta<\nu_{0}<\nu_{1}$ and $\gamma_{0}<\zeta<\gamma$ as in (B), (C).

Now, by Lemma 2.3 and (2), we have ${\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi(\mathbf{c}_{0},\mathbf{c})$. We further get $\boldsymbol{h}{\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\boldsymbol{h}\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi(\boldsymbol{h}\mathbf{c}_{0},\boldsymbol{h}\mathbf{c})$ by Theorem 5.1 because $\varphi$ is a $\mathscr{L}\varPi^{1}_{n+1}$ formula and $\boldsymbol{h}$ belongs to $\text{\rm PERM}_{m-1}$. However $\boldsymbol{h}\mathbf{c}_{0}=\mathbf{c}_{0}$ since $|\mathbf{c}_{0}|\cap|\boldsymbol{h}|=\varnothing$ by (B). Thus ${\boldsymbol{p}}_{0}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi(\mathbf{c}_{0},\mathbf{c}^{\prime})$ holds by Theorem 4.1 and (C), where $\mathbf{c}^{\prime}=\boldsymbol{h}\mathbf{c}$, ${\boldsymbol{p}}_{0}^{\prime}=\boldsymbol{h}{\boldsymbol{p}}_{0}$.

Note that the common part $|{\boldsymbol{p}}_{0}|\cap|{\boldsymbol{p}}_{0}^{\prime}|$ of the domains of ${\boldsymbol{p}}_{0},{\boldsymbol{p}}_{0}^{\prime}$ does not intersect $|\boldsymbol{h}|$ by (B) since $|{\boldsymbol{p}}_{0}|\subseteq\delta\times\omega$ by (3). It follows that ${\boldsymbol{p}}_{0},{\boldsymbol{p}}_{0}^{\prime}$ are compatible, basically ${\boldsymbol{p}}={\boldsymbol{p}}_{0}\cup{\boldsymbol{p}}_{0}^{\prime}$ is a multitree in $\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)$. Thus ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}^{\prime}$ and still ${\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}\varphi(\mathbf{c}_{0},\mathbf{c}^{\prime})$. It follows by Lemma 6.3 that

1. (5)

${\boldsymbol{p}}$  ${\boldsymbol{\mathbb{P}}}$-forces $\varphi(\mathbf{c}_{0}[{\underline{G}}],\mathbf{c}^{\prime}[{\underline{G}}])$ over ${\mathbf{L}}$.

However $|\mathbf{c}^{\prime}|\subseteq\omega_{1}\times m$ by construction because $|\mathbf{c}|\subseteq\delta\times\omega$ by (3), and hence $\mathbf{c}^{\prime}[{\underline{G}}]\in{\mathbf{L}}[{{\underline{G}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$ is forced. Thus ${\boldsymbol{p}}$  ${\boldsymbol{\mathbb{P}}}$-forces ${\exists\,}x\in{\mathbf{L}}[{{\underline{G}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]\,\varphi(\mathbf{c}_{0}[{\underline{G}}],x)$ over ${\mathbf{L}}$ by (5), contrary to (1). The contradiction ends Case 1.

Case 2: negative decision, no multisequence in $U$ extends ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$. We can assume that $\gamma>\gamma_{0}$. (Otherwise replace $\gamma$ by $\gamma_{0}+1$.) Let $\nu_{0}$ be the lest ordinal, bigger than $\delta$ and satisfying $|\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma|\subseteq\nu_{0}\times\omega$. Let $\nu_{1}=\nu_{0}+\omega$. Then countable sets $D=\delta\times[m,\infty)$ and $R$ as in (B) are defined and $D\cap R=\varnothing$, so we can fix a transformation $\boldsymbol{h}\in\text{\rm PERM}_{m-1}$ satisfying (B). Note that $D\subseteq\delta\times\omega\subseteq|\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma|$ by (4) but $R\cap|\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma|=\varnothing$ by the choice of $\nu_{0}$.

Pick $\lambda<\omega_{1}$ such that $\lambda>\gamma>\gamma_{0}$. Then the multisequence $\vv\text{\leavevmode{}}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\lambda$ clearly satisfies (A), (B) and extends $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$. Our plan is now to slightly modify $\vv\text{\leavevmode{}}$ in order to fulfill (C) as well, with $\zeta=\gamma$. Such a minor modification consists in the replacement of the ${R}$-part of $\vv\text{\leavevmode{}}$ above $\gamma$ by the ${\boldsymbol{h}}$-copy of its ${D}$-part.

To present this in detail, recall that $\vv\text{\leavevmode{}}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\lambda=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\lambda}$, where each $\nmi\Pi_{\alpha}$ is a small multiforcing, whose domain $d_{\alpha}=|\nmi\Pi_{\alpha}|\subseteq\omega_{1}\times\omega$ is countable. If $\alpha<\gamma$ then put $\boldsymbol{\pi}_{\alpha}=\nmi\Pi_{\alpha}$. Suppose that $\gamma\leq\alpha<\lambda$. Then $D\subseteq|\nmi\Pi_{\alpha}|$ by (4). Define a modified multiforcing $\boldsymbol{\pi}_{\alpha}$ such that

1. (a)

$|\boldsymbol{\pi}_{\alpha}|=d_{\alpha}\cup R$ — note that $D\subseteq d_{\alpha}\subseteq|\boldsymbol{\pi}_{\alpha}|$ in this case because $D\subseteq|\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma|$ by (4) (as $\gamma_{0}\leq\gamma$), and hence $D\subseteq d_{\alpha}=|\nmi\Pi_{\alpha}|$ (as $\alpha\geq\gamma$), 2. (b)

if $\langle\xi,k\rangle\in d_{\alpha}\smallsetminus R$ then $\boldsymbol{\pi}_{\alpha}(\xi,k)=\nmi\Pi_{\alpha}(\xi,k)$, 3. (c)

if $\langle\xi,k\rangle\in D$, so $\boldsymbol{h}(\xi,k)=\langle\eta,m-1\rangle\in R$, then $\boldsymbol{\pi}_{\alpha}(\eta,m-1)=\nmi\Pi_{\alpha}(\xi,k)$.

We claim that $\vv\boldsymbol{\pi}=\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ is a multisequence, that is, if $\alpha<\beta<\lambda$ then $\boldsymbol{\pi}_{\alpha}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\pi}_{\beta}$. This amounts to the folowing: if $\langle\eta,k\rangle\in|\boldsymbol{\pi}_{\alpha}|$ then $\boldsymbol{\pi}_{\alpha}(\eta,k)\sqsubset\boldsymbol{\pi}_{\beta}(\eta,k)$. Note that $\boldsymbol{\pi}_{\alpha}(\eta,k)=\nmi\Pi_{\alpha}(\eta,k)$ in case $\langle\eta,k\rangle\notin R$.

Thus it remains to check that $\boldsymbol{\pi}_{\alpha}(\eta,m-1)\sqsubset\boldsymbol{\pi}_{\beta}(\eta,m-1)$ whenever $\alpha<\beta<\lambda$, $\langle\eta,m-1\rangle=\boldsymbol{h}(\xi,k)\in R\cap|\boldsymbol{\pi}_{\alpha}|$, and $\langle\xi,k\rangle\in D$. If now $\alpha<\gamma$ then $R\cap|\boldsymbol{\pi}_{\alpha}|=\varnothing$ by the choice of $\nu_{0}$, so it remains to consider the case when $\gamma\leq\alpha$. Then the pairs $\langle\xi,k\rangle$, $\langle\eta,m-1\rangle$ belong to $|\boldsymbol{\pi}_{\alpha}|$ by construction, and we have $\boldsymbol{\pi}_{\alpha}(\eta,m-1)=\nmi\Pi_{\alpha}(\xi,k)$ and $\boldsymbol{\pi}_{\beta}(\eta,m-1)=\nmi\Pi_{\beta}(\xi,k)$. Therefore $\boldsymbol{\pi}_{\alpha}(\xi,m)\sqsubset\boldsymbol{\pi}_{\beta}(\xi,m)$ since $\vv{\nmi\Pi}$ is a multisequence, and we are done.

Now we claim that the multisequence $\vv\boldsymbol{\pi}=\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ satisfies (A), (B), (C). Indeed as the difference between each $\boldsymbol{\pi}_{\alpha}$ and the corresponding $\nmi\Pi_{\alpha}$ is fully located in the domain $R=\{\hskip 0.43057pt{\langle\xi,m-1\rangle\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\nu_{0}\leq\xi<\nu_{1}\hskip 0.43057pt\}$, we have ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-1}={\vv\text{\leavevmode{}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm-1}$, therefore $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$. We also note that $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma=\vv\text{\leavevmode{}}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ by construction, hence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma=\vv\text{\leavevmode{}}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset\vv\boldsymbol{\pi}$. This implies (A).

We also have (B) by construction. We finally claim that (C) is satisfied with $\zeta=\gamma$, that is, if $\gamma\leq\alpha<\lambda$ then $\boldsymbol{h}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}\boldsymbol{\pi}_{\alpha}=\boldsymbol{\pi}_{\alpha}$. Indeed we have $D\cup R\subseteq|\boldsymbol{\pi}_{\alpha}|$, see (a). Now the invariance of $\boldsymbol{\pi}_{\alpha}$ under $\boldsymbol{h}$ holds by (b), (c).

It follows that ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in U$. In addition, ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ extends ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, since $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset\vv\boldsymbol{\pi}$. But this contradicts the Case 2 assumption.

To conclude, either case leads to a contradiction, proving the theorem. ∎

4 Non-wellorderability

We finally prove that the reals are not wellorderable by a (lightface) analytically definable relation in ${\boldsymbol{\mathbb{P}}}$-generic extensions, that is, (IV) of Theorem 1.1.

Theorem 4.1**.**

Assume that $m<\omega$ and a set $G\subseteq\boldsymbol{\mathbb{P}}$ is ${\boldsymbol{\mathbb{P}}}$-generic over ${\mathbf{L}}$. Then it is true in ${\mathbf{L}}[G]$ that the reals are not wellorderable by an analytically definable relation.

Proof**.**

Suppose to the contrary that, in ${\mathbf{L}}[G]$, a $\varSigma^{1}_{m+2}$ relation $\ll$ strictly wellorders $\omega^{\omega}$, $m\geq 1$. Let $\psi(x,y)$ be a parameter-free $\varSigma^{1}_{m+2}$ formula, which defines $\ll$, so that $x\ll y$ iff $\psi(x,y)$ in ${\mathbf{L}}[G]$. Note that $\ll$ is essentially a $\varDelta^{1}_{m+2}$ relation, since $x\ll y\,\mathbin{\,\Longleftrightarrow\,}\,y\not\ll x\land x\neq y$.

Of all nonconstructible reals $x_{\xi m}[G]$, $\xi<\omega_{1}$, there is a ${\ll}$-least one. We suppose that $x_{0m}[G]$ is such. (If it is some $x_{\xi_{0}m}[G]$, $\xi_{0}\neq 0$, then the arguments suitably change in obvious way.) That is, $x_{0m}[G]\ll x_{\xi m}[G]$ whenever $\xi>0$. Accordingly there is a multitree ${\boldsymbol{p}}_{0}\in G$ that ${\boldsymbol{\mathbb{P}}}$-forces, over ${\mathbf{L}}$, that

(1) $\ll$ (that is, the relation defined by $\psi$) is a wellordering of $\omega^{\omega}$, and

(2) ${\forall\,}\xi>0\,(x_{0m}[{\underline{G}}]\ll x_{\xi m}[{\underline{G}}])$.

Therefore, if $\xi>0$ then ${\boldsymbol{p}}_{0}$ ${\boldsymbol{\mathbb{P}}}$-forces {\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m}[{\underline{G}}]\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m}[{\underline{G}}]) over ${\mathbf{L}}$. (We make use of the real names {\overset{\text{\hskip 2.15277pt{}_{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k} introduced by 1.6, 1.7.)

By Lemma 6.3, we can assume that ${\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}(1)\land(2)$, so that in fact we have ${\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma_{0}}\hskip 1.72218pt}(1)\land(2)$, for some $\gamma_{0}<\omega_{1}$. Then ${\boldsymbol{p}}_{0}\in\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{0})=\text{MT}(\nmi\Pi_{<\gamma_{0}})$, where $\nmi\Pi_{<\gamma_{0}}=\bigcup^{\text{\tt cw}}_{\xi<\gamma_{0}}\nmi\Pi_{\xi}$ is a small multiforcing. Let $\delta<\omega_{1}$ be the least ordinal satisfying $|\nmi\Pi_{<\gamma_{0}}|\subseteq\delta\times\omega$. It follows then that $|{\boldsymbol{p}}_{0}|\subseteq\delta\times\omega$.

By Lemma 6.4, there is an ordinal $\gamma_{1}$, $\gamma_{0}<\gamma_{1}<\omega_{1}$, such that if $\xi<\omega_{1}$ then {\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma_{1}}\hskip 1.72218pt}({\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m}\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m})^{-}. We can enlarge $\gamma_{1}$, if necessary, using Lemma 5.2, to make sure that $\langle 0,m\rangle\in|\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{1}|$, that is, $\langle 0,m\rangle\in|\nmi\Pi_{\alpha^{\prime}}|$ for some $\alpha^{\prime}<\gamma_{1}$.

If $\xi<\omega_{1}$ then let $\boldsymbol{h}_{\xi}\in\text{\rm PERM}_{m}$ be the permutation of $\langle 0,m\rangle$ and $\langle\xi,m\rangle$, such that $|\boldsymbol{h}_{\xi}|=\{\hskip 0.43057pt\langle 0,m\rangle,\langle\xi,m\rangle\hskip 0.43057pt\}$, $\boldsymbol{h}_{\xi}(0,m)=\langle\xi,m\rangle$, $\boldsymbol{h}_{\xi}(\xi,m)=\langle 0,m\rangle$, $\boldsymbol{h}_{\xi}(\eta,n)=\langle\eta,n\rangle$ for any pair $\langle\eta,n\rangle$ different from both $\langle 0,m\rangle$ and $\langle\xi,m\rangle$.

The remainder of the proof is very similar to the proof of Theorem 3.1. Let $U$ consist of all multisequences of the form ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, where

1. (A)

$\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$, $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{1}\subset\vv\boldsymbol{\pi}$, and hence ${\boldsymbol{p}}_{0}\in\text{MT}(\vv\boldsymbol{\pi})$ by (2);

and there exist ordinals $\xi,\zeta<\omega_{1}$ such that

1. (B)

$\delta<\xi<\mathop{\tt dom}\vv\boldsymbol{\pi}$; 2. (C)

$\gamma_{1}\leq\zeta<\mathop{\tt dom}\vv\boldsymbol{\pi}$ and $(\boldsymbol{h}_{\xi}\vv\boldsymbol{\pi}){{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\zeta}=\vv\boldsymbol{\pi}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}_{\geq\zeta}$, or equivalently $\boldsymbol{h}_{\xi}{\hskip 0.86108pt\boldsymbol{\cdot}\hskip 0.86108pt}(\vv\boldsymbol{\pi}(\alpha))=\vv\boldsymbol{\pi}(\alpha)$ whenever $\zeta\leq\alpha<\mathop{\tt dom}\vv\boldsymbol{\pi}$.

It follows from Lemma 1.2 that $U$ is a ${\mathbf{\Sigma}}^{\mathrm{HC}}_{m+1}$ set (with $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{1}$ as a parameter). Therefore by 4.4(ii) there is an ordinal $\gamma<\omega_{1}$ such that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$  ${m}$-decides $U$.

Case 1: ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in U$. Basically this means that $\gamma>\gamma_{1}$ and there are ordinals $\xi<\omega_{1}$ and $\zeta<\gamma$ such that (A), (B), (C) hold for $\xi$ and the multisequence $\vv\boldsymbol{\pi}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$. By Lemma 2.3 and the choice of $\gamma_{1}$, {\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}({\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m}\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m})^{-} holds. This implies \boldsymbol{h}_{\xi}{\boldsymbol{p}}_{0}\mathop{\hskip 1.72218pt{\tt forc}_{\boldsymbol{h}_{\xi}\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}({\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m}\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m})^{-} by Theorem 5.1 because ({\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m}\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m})^{-} is a $\mathscr{L}\varPi^{1}_{m+2}$ formula and $\boldsymbol{h}_{\xi}$ belongs to $\text{\rm PERM}_{m}$. We conclude that {\boldsymbol{p}}_{0}^{\prime}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}({\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m}\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m})^{-} by (C) and Theorem 4.1, where ${\boldsymbol{p}}_{0}^{\prime}=\boldsymbol{h}_{\xi}{\boldsymbol{p}}_{0}$.

Note that $\langle\xi,m\rangle\notin|{\boldsymbol{p}}_{0}|$ by (B). It follows by the definition of $\boldsymbol{h}_{\xi}$ that ${\boldsymbol{p}}_{0},{\boldsymbol{p}}_{0}^{\prime}$ are compatible, basically ${\boldsymbol{p}}={\boldsymbol{p}}_{0}\cup{\boldsymbol{p}}_{0}^{\prime}$ is a multitree in $\text{MT}(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)$ and ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}$, ${\boldsymbol{p}}\leqslant{\boldsymbol{p}}_{0}^{\prime}$. Thus both {\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}({\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m}\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m})^{-} and {\boldsymbol{p}}\mathop{\hskip 1.72218pt{\tt forc}_{\vv{\nmi\Pi}{\hskip 0.60275pt{\restriction}\hskip 0.60275pt}\gamma}\hskip 1.72218pt}({\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi m}\ll{\overset{\text{\hskip 2.15277pt{}{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{0m})^{-} hold, contrary to Lemma 6.2(iv). The contradiction ends Case 1.

Case 2: negative decision, no multisequence in $U$ extends ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$. We can assume that $\gamma>\gamma_{1}$ (otherwise replace $\gamma$ by $\gamma_{1}+1$). Note that $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\gamma}$, where each $\nmi\Pi_{\alpha}$ is a small multiforcing, whose domain $d_{\alpha}=|\nmi\Pi_{\alpha}|\subseteq\omega_{1}\times\omega$ is countable. Therefore the set $d=|\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma|=\bigcup_{\alpha<\gamma}d_{\alpha}$ is still countable. Pick an ordinal $\xi$, $\delta<\xi<\omega_{1}$, such that $\langle\xi,m\rangle\notin d$. Finally pick an ordinal $\lambda$, $\gamma<\lambda<\omega_{1}$. Then $\vv\text{\leavevmode{}}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\lambda$ (as $\vv\boldsymbol{\pi}$) and $\xi$ clearly satisfy (A) and (B), and $\vv\text{\leavevmode{}}$ extends $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$. Let’s somewhat modify $\vv\text{\leavevmode{}}$ in order to fulfill (C) as well.

As above, $\vv\text{\leavevmode{}}=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\lambda=\langle\nmi\Pi_{\alpha}\rangle_{\alpha<\lambda}$, each $\nmi\Pi_{\alpha}$ is a small multiforcing, and its domain $d_{\alpha}=|\nmi\Pi_{\alpha}|\subseteq\omega_{1}\times\omega$ is countable. If $\alpha<\gamma$ then put $\boldsymbol{\pi}_{\alpha}=\nmi\Pi_{\alpha}$. Suppose that $\gamma\leq\alpha<\lambda$. Then $\alpha\geq\gamma_{1}$, and hence $\langle 0,m\rangle\in|\nmi\Pi_{\alpha}|$ by the choice of $\gamma_{1}$. Define a modified multiforcing $\boldsymbol{\pi}_{\alpha}$ such that $|\boldsymbol{\pi}_{\alpha}|=d_{\alpha}\cup\{\hskip 0.43057pt\langle\xi,m\rangle\hskip 0.43057pt\}$, if $\langle\eta,k\rangle\in d_{\alpha}\smallsetminus\{\hskip 0.43057pt\langle\xi,m\rangle\hskip 0.43057pt\}$ then $\boldsymbol{\pi}_{\alpha}(\eta,k)=\nmi\Pi_{\alpha}(\eta,k)$, and finally $\boldsymbol{\pi}_{\alpha}(\xi,m)=\nmi\Pi_{\alpha}(0,m)$.

We claim that $\vv\boldsymbol{\pi}=\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ is a multisequence, that is, if $\alpha<\beta<\lambda$ then $\boldsymbol{\pi}_{\alpha}\mathrel{\boldsymbol{\sqsubset}}\boldsymbol{\pi}_{\beta}$. This amounts to the folowing: if $\langle\eta,k\rangle\in|\boldsymbol{\pi}_{\alpha}|$ then $\boldsymbol{\pi}_{\alpha}(\eta,k)\sqsubset\boldsymbol{\pi}_{\beta}(\eta,k)$. Note that $\boldsymbol{\pi}_{\alpha}(\eta,k)=\nmi\Pi_{\alpha}(\eta,k)$ whenever $\langle\eta,k\rangle\neq\langle\xi,m\rangle$. Thus it remains to check that $\boldsymbol{\pi}_{\alpha}(\xi,m)\sqsubset\boldsymbol{\pi}_{\beta}(\xi,m)$ given $\alpha<\beta<\lambda$ such that $\langle\xi,m\rangle\in|\boldsymbol{\pi}_{\alpha}|$. If $\alpha<\gamma$ then $|\boldsymbol{\pi}_{\alpha}|=|\nmi\Pi_{\alpha}|=d_{\alpha}$ by construction, and hence $\langle\xi,m\rangle\notin d_{\alpha}$ by the choice of $\xi$. It remains to consider the case $\gamma\leq\alpha<\lambda$. Then $\langle 0,m\rangle\in d_{\alpha}$ (see above), hence the pairs $\langle 0,m\rangle$, $\langle\xi,m\rangle$ belong to $|\boldsymbol{\pi}_{\alpha}|$ by construction, and then obviously belong to $|\boldsymbol{\pi}_{\beta}|$ as $\alpha<\beta$. Now $\boldsymbol{\pi}_{\alpha}(\xi,m)=\nmi\Pi_{\alpha}(0,m)$ and $\boldsymbol{\pi}_{\beta}(\xi,m)=\nmi\Pi_{\beta}(0,m)$, and we have $\boldsymbol{\pi}_{\alpha}(\xi,m)\sqsubset\boldsymbol{\pi}_{\beta}(\xi,m)$ since $\vv{\nmi\Pi}$ is a multisequence.

Now we claim that the multisequence $\vv\boldsymbol{\pi}=\langle\boldsymbol{\pi}_{\alpha}\rangle_{\alpha<\lambda}$ satisfies (A), (B), (C) with $\zeta=\gamma$. If $\alpha<\lambda$ then the difference between $\nmi\Pi_{\alpha}$ and $\boldsymbol{\pi}_{\alpha}$ is located in the one-element domain $\{\hskip 0.43057pt\langle\xi,m\rangle\hskip 0.43057pt\}$, therefore ${\boldsymbol{\pi}_{\alpha}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}={\nmi\Pi_{\alpha}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$. It follows that ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}={(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\lambda)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$, hence $\vv\boldsymbol{\pi}\in\vv\text{sMF}[{\vv{\nmi\Pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]$. We further have $\vv\boldsymbol{\pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma=\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma$ by construction. Thus $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset\vv\boldsymbol{\pi}$, hence $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma_{1}\subset\vv\boldsymbol{\pi}$, and we have (A).

We also have (B) and (C) (with $\zeta=\gamma$) by construction.

Thus ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}\in U$. In addition, ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ extends ${(\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma)}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$, since even more $\vv{\nmi\Pi}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}\gamma\subset\vv\boldsymbol{\pi}$ by construction. But this contradicts to the Case 2 assumption.

To conclude, either case leads to a contradiction, proving the theorem. ∎

5 Proof of the main theorem

Proof** (Theorem 1.1).**

We consider a ${\boldsymbol{\mathbb{P}}}$-generic extension ${\mathbf{L}}[G]$ of ${\mathbf{L}}$ and present it in the form ${\mathbf{L}}[G]={\mathbf{L}}[X]$ as in Section 1, where $X=\langle x_{\xi k}\rangle_{\langle\xi,k\rangle\in\omega_{1}\times\omega}$, and each $x_{\xi k}=x_{\xi k}[G]$ is a real in $2^{\omega}\cap{\mathbf{L}}[G]$. We also consider the subextensions ${\mathbf{L}}[{G}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}]={\mathbf{L}}[\langle x_{\xi k}\rangle_{\xi<\omega_{1}\land k<m}]$ of ${\mathbf{L}}[G]={\mathbf{L}}[X]$. Then (I) and (II) of Theorem 1.1 hold by Corollary 2.3, (III) holds by Theorem 3.1, and finally (IV) holds by Theorem 4.1. ∎

Index

• a. d. (trees) Definition 1.1

• antichain Definition 2.4

• avoids Definition 4.2

• base, $\mathop{\tt base}({{\mathbb{P}}})$ item special,

• canonical homeomorphism

• $\boldsymbol{h}_{ST}$ §1

• $\boldsymbol{h}_{T}$ §1

• Cohen forcing, ${{\mathbb{P}}}_{\text{\rm coh}}$ Example 2.1

• componentwise union

• $\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}}$ Definition 4.4

• $\bigcup^{\text{\tt cw}}\vv\boldsymbol{\pi}=\bigcup^{\text{\tt cw}}_{\alpha<\lambda}{\boldsymbol{\pi}}_{\alpha}$ Definition 4.4

• CTM, countable transitive model Definition 1.3

• decision

• ${m}$-decides Definition 2.1

• negative Definition 2.1

• positive Definition 2.1

• dense item $-$, item $-$

• diamond sequences

• ${\vv\boldsymbol{\pi}}\lceil{\mu,m}\rceil$, $\mathord{\lceil\mu,m\rceil}^{+}$ item (III)

• ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil\hskip 0.86108pt,\linebreak[0]\>{D}\lceil{\mu}\rceil\hskip 0.86108pt,\linebreak[0]\>{z}\lceil{\mu}\rceil$ Theorem 3.1

• ${m}\lceil{\mu}\rceil$, ${p}\lceil{\mu}\rceil$, ${\delta}\lceil{\mu}\rceil$ item (II)

• directly forces 1st item

• domain-continuous Definition 1.1

• finite splitting system, FSS Definition 3.1

• empty system, $\boldsymbol{\Lambda}$ Definition 3.1

• extension proper, $\psi\prec\varphi$ §3

• extension, $\psi\preccurlyeq\varphi$ §3

• height, $\text{\tt hgt}(\varphi)$ Definition 3.1

• over ${{\mathbb{P}}}$, $\text{\bf FSS}({{\mathbb{P}}})$ §3

• tree occurs in §3

• forcing

• $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$ item $1^{\circ}$a

• $\mathop{\hskip 1.72218pt{\tt wforc}\hskip 1.72218pt}$ item $1^{\circ}$b

• formula

• ${\mathscr{L}}$-formula §1

• $\mathscr{L}\varSigma^{1}_{n}$, $\mathscr{L}\varPi^{1}_{n}$, $\mathscr{L}(\varSigma{+}\varPi)^{1}_{n}$ §1

• $\varphi^{-}$ §1

• generic

• ${{{\mathbb{P}}}}$-generic real item (ii)

• key elements

• ${{\mathbb{C}}}$ Proposition 4.3

• ${{\mathbb{C}}}^{\prime}$ Corollary 5.5

• key maps $\mu_{m}$ Proof

• $\mu_{m}(\gamma)$ Proof

• $\boldsymbol{\mathbb{P}}$ Definition 5.1

• $\boldsymbol{\mathbb{P}}_{<\gamma}$ ∎ ‣ Definition 5.1

• $\boldsymbol{\mathbb{P}}_{\geq\gamma}$ §5

• $\nmi\Pi$ Definition 5.1

• $\vv{\nmi\Pi}$ Proof

• $\nmi\Pi_{\alpha}$ Proof

• $\nmi\Pi_{<\gamma}$ Definition 5.1

• $\nmi\Pi_{\geq\gamma}$ Definition 5.1

• layer restriction

• $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• length

• $\mathop{\tt dom}(\vv\boldsymbol{\pi})$ Definition 1.1

• locks Definition 1.3, Definition 2.2, Definition 3.1, Definition 4.2

• model

• CTM, countable transitive model Definition 1.3

• ${{\mathfrak{M}}}(x)$ Definition 1.3

• multiforcing §4

• extension, $\vv\boldsymbol{\pi}\subset\vv\text{\leavevmode{}}$ 2nd item

• ${{{\mathfrak{M}}}}$-extension, $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$ 2nd item

• sMF §1

• $\boldsymbol{\mathbb{P}}$ Definition 5.1

• $\boldsymbol{\mathbb{P}}_{<\gamma}$ ∎ ‣ Definition 5.1

• $\boldsymbol{\mathbb{P}}_{\geq\gamma}$ §5

• $\nmi\Pi$ Definition 5.1

• $\vv{\nmi\Pi}$ Proof

• $\nmi\Pi_{\alpha}$ Proof

• $\nmi\Pi_{<\gamma}$ Definition 5.1

• $\nmi\Pi_{\geq\gamma}$ Definition 5.1

• refinement

• generic Definition 3.1

• refinement, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$ §2

• refinement, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}$ Definition 3.1

• refinement, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$ Definition 2.2

• refinement, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$ Definition 4.2

• refinement, $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}$ §7

• regular item $-$

• small item $-$

• special item $-$

• multisequence Definition 1.1

• key multisequence Definition 4.4

• multisystem Definition 5.1

• 2wise disjoint §5

• extension, $\boldsymbol{\psi}\preccurlyeq\boldsymbol{\varphi}$ §5

• $h^{\boldsymbol{\varphi}}(\xi,k,m)$ item (1)

• $\text{MS}(\boldsymbol{\pi})$ Definition 5.1

• ${\boldsymbol{\pi}}$-multisystem Definition 5.1

• $T^{\boldsymbol{\varphi}}_{\xi k,m}(s)$ item (2)

• multitree §4

• disjoint union, ${\boldsymbol{p}}\cup{\boldsymbol{q}}$ footnote 3

• empty multitree, $\boldsymbol{\Lambda}$ §4

• MT §4

• $\text{MT}(\vv\boldsymbol{\pi})$ Definition 1.1

• $\text{MT}(\boldsymbol{\pi})$ §4

• ${\boldsymbol{\pi}}$-multitree §4

• restriction, ${\boldsymbol{p}}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}X$ Proof

• somewhere almost disjoint, s. a. d. §4

• ] §4

• $T^{{\boldsymbol{p}}}_{\xi k}$ §4

• occurs §3

• open item $-$, item $-$

• perfect-tree forcing, PTF §2

• base, $\mathop{\tt base}({{\mathbb{P}}})$ item special,

• refinement, ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}$ Definition 1.1

• refinement, ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{Q}}}$ Definition 1.3

• regular item regular,

• small item small,

• special item special,

• permutation

• action §5

• PERM §5

• ${{{\mathbb{P}}}}$-generic item (ii)

• pre-dense item $-$, item $-$

• principal generic reals, $x_{\xi k}[G]$ §4

• real

• ${{{\mathbb{P}}}}$-generic item (ii)

• real name Definition 1.1

• evaluation, $\mathbf{c}[G]$ Definition 1.2

• ${\vv\boldsymbol{\pi}}$-real name Definition 1.1

• true Definition 1.1

• ${\boldsymbol{\pi}}$-real name Definition 1.3

• true §1

• small Definition 1.1

• true ${\boldsymbol{\pi}}$-real name §1

• {\overset{\text{\hskip 2.15277pt{}_{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k} Example 1.6

• reals

• principal generic reals, $x_{\xi k}[G]$ §4

• refinement

• generic Definition 3.1

• locks Definition 1.3

• refinement, ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{Q}}}$ Definition 1.3

• refinement, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}$ Definition 3.1

• refinement, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$ Definition 2.2

• refinement, $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$ Definition 4.2

• refinement, $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}$ §7

• regular item regular,, item $-$

• s. a. d., somewhere a. d. §4

• set

• ${{\mathbb{C}}}$ Proposition 4.3

• ${{\mathbb{C}}}^{\prime}$ Corollary 5.5

• set of multitrees

• dense item $-$

• open item $-$

• pre-dense item $-$

• set of trees

• dense item $-$

• open item $-$

• pre-dense item $-$

• sets $\Upsilon_{m}(p)$ Proof

• small item small,, item $-$

• s. a. d., somewhere a. d. §4

• somewhere almost disjoint, s. a. d. §4

• special item special,, item $-$

• splitting §1

• stem, $\text{\tt stem}(T)$ §1

• strings §1

• empty string, $\Lambda$ §1

• ${\gamma}$-tail §4

• tree §1

• $[s]$ Example 2.1

• almost disjoint Definition 1.1

• ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}$ Definition 3.1

• ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(s)$ Definition 3.1

• $T^{{\boldsymbol{p}}}_{\xi k}$ §4

• $\mathop{\tt tree}(X)$ Definition 1.1

• universal formula, $\text{un}_{m}(p,x)$ Proof

• $\leqslant_{{\mathbf{L}}}$ Proof

• $2^{n}$ §1

• $\mathop{\tt base}({{\mathbb{P}}})$ item special,

• ${{\mathbb{C}}}$ Proposition 4.3

• ${{\mathbb{C}}}^{\prime}$ Corollary 5.5

• $\mathbf{c}[G]$ Definition 1.2

• $|\mathbf{c}|$ Definition 1.1

• ${D}\lceil{\mu}\rceil$ Theorem 3.1

• ${D}^{|\boldsymbol{u}|}_{{\boldsymbol{q}}}$ ∎ ‣ item $(*)$

• $\mathop{\hskip 1.72218pt{\tt forc}\hskip 1.72218pt}$ item $1^{\circ}$a

• FSS Definition 3.1

• $\text{\bf FSS}({{\mathbb{P}}})$ §3

• $\mathrm{HC}$ Proof

• $h^{\boldsymbol{\varphi}}(\xi,k,m)$ item (1)

• $\boldsymbol{h}_{ST}$ §1

• $\boldsymbol{h}_{T}$ §1

• $K^{\mathbf{c}}_{n}$ item $(†)$

• $K^{\mathbf{c}}_{ni}$ Definition 1.1

• $\Lambda$ §1

• $\mathop{\tt dom}(\vv\boldsymbol{\pi})$ Definition 1.1

• ${\mathscr{L}}$-formula §1

• $\mathop{\tt lh}(s)$ §1

• $\mathscr{L}\varSigma^{1}_{n}$, $\mathscr{L}\varPi^{1}_{n}$, $\mathscr{L}(\varSigma{+}\varPi)^{1}_{n}$ §1

• sMF §1

• $\vv\text{sMF}$ Definition 1.1

• $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}_{\omega_{1}}$ Definition 1.1

• $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\vv\text{sMF}_{\omega_{1}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\text{sMF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{MF}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• ${m}\lceil{\mu}\rceil$ item (II)

• $\text{MS}(\boldsymbol{\pi})$ Definition 5.1

• MT §4

• $\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\text{MT}{}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\text{MT}(\boldsymbol{\pi})$ §4

• $\text{MT}(\vv\boldsymbol{\pi})$ Definition 1.1

• $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• $\text{MT}(\boldsymbol{\pi}){}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\mathord{\lceil\mu,m\rceil}^{+}$ item (III)

• $\mu_{m}(\gamma)$ Proof

• ${{\mathfrak{M}}}(x)$ Definition 1.3

• ] §4

• ${{\mathbb{P}}}_{\text{\rm coh}}$ Example 2.1

• $\boldsymbol{\mathbb{P}}$ Definition 5.1

• $\boldsymbol{\mathbb{P}}_{<\gamma}$ ∎ ‣ Definition 5.1

• $\boldsymbol{\mathbb{P}}_{\geq\gamma}$ §5

• PERM §5

• $\varphi^{-}$ §1

• $\nmi\Pi$ Definition 5.1

• $\vv{\nmi\Pi}$ Proof

• $\nmi\Pi_{\alpha}$ Proof

• ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• ${\vv\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• $\nmi\Pi_{<\gamma}$ Definition 5.1

• ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• $\nmi\Pi_{\geq\gamma}$ Definition 5.1

• ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{<}\hskip 0.60275ptm}$ §2

• ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{{\geq}\hskip 0.60275ptm}$ §2

• ${\boldsymbol{\pi}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• ${{\boldsymbol{p}}}{}\mathord{{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}{{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}}}_{\hskip 0.60275ptm}$ §2

• ${\vv\boldsymbol{\pi}}\lceil{\mu,m}\rceil$ item (III)

• ${\vv\boldsymbol{\pi}}\lceil{\mu}\rceil$ Theorem 3.1

• $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{\mathbf{c}}\text{\leavevmode{}}$ Definition 3.1

• $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}_{{\boldsymbol{D}}}\text{\leavevmode{}}$ Definition 2.2

• $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}^{\mathbf{c}}_{\xi k}\text{\leavevmode{}}$ Definition 4.2

• $\boldsymbol{\pi}\mathrel{{\mathrel{\boldsymbol{\sqsubset}}}{\mathrel{\boldsymbol{\sqsubset}}}}_{{{\mathfrak{M}}}}\text{\leavevmode{}}$ §7

• $\boldsymbol{\pi}\mathrel{\boldsymbol{\sqsubset}}\text{\leavevmode{}}$ §2

• $\vv\boldsymbol{\pi}\subset_{{\mathfrak{M}}}\vv\text{\leavevmode{}}$ 2nd item

• $\vv\boldsymbol{\pi}\subset\vv\text{\leavevmode{}}$ 2nd item

• ${\boldsymbol{p}}{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}X$ Proof

• ${p}\lceil{\mu}\rceil$ item (II)

• ${{\mathbb{P}}}\sqsubset_{D}{{\mathbb{Q}}}$ Definition 1.3

• ${{\mathbb{P}}}\sqsubset{{\mathbb{Q}}}$ Definition 1.1

• PT §1

• PTF §2

• ${\boldsymbol{p}}\cup{\boldsymbol{q}}$ footnote 3

• ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}$ Definition 3.1

• ${\boldsymbol{Q}}^{\mathbb{\Phi}}_{\xi k,m}(s)$ Definition 3.1

• ${\delta}\lceil{\mu}\rceil$ item (II)

• ${{{\mathbb{Q}}}}^{\mathbb{\Phi}}_{\xi k}$ Definition 3.1

• ] Example 2.1

• $\subseteq^{\text{\tt fin}}$ §1, §2

• $\subseteq^{\text{\tt fin}}\bigvee$ §2

• $s\subseteq t$ §1

• $s\subset t$ §1

• $\subset_{{\mathfrak{M}}}$ 2nd item

• $T{(\to i)}$ §1

• $T{(\to u)}$ §1

• ] Definition 1.1

• $t{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}k$ §1

• $T^{\boldsymbol{\varphi}}_{\xi k,m}(s)$ item (2)

• $T{\hskip 0.86108pt{\restriction}\hskip 0.86108pt}_{s}$ §1

• $T^{{\boldsymbol{p}}}_{\xi k}$ §4

• $\mathop{\tt tree}(X)$ Definition 1.1

• $\boldsymbol{\pi}\cup^{\text{\tt cw}}\text{\leavevmode{}}$ Definition 4.4

• $\bigcup^{\text{\tt cw}}_{\alpha<\lambda}{\boldsymbol{\pi}}_{\alpha}$ Definition 4.4

• $\mathop{\hskip 1.72218pt{\tt wforc}\hskip 1.72218pt}$ item $1^{\circ}$b

• {\overset{\text{\hskip 2.15277pt{}_{\text{\Large\bf.}}}}{\boldsymbol{x}}}_{\xi k} Example 1.6

• $x_{\xi k}[G]$ Remark 4.3

• $\Upsilon_{m}(p)$ Proof

• ${z}\lceil{\mu}\rceil$ Theorem 3.1