Definable ${\mathsf E}_0$ classes at arbitrary projective levels

TL;DR

This paper constructs a model of ZFC where, for any given projective level n≥3, there exists a non-OD E_0 equivalence class that is a countable, lightface Pi^1_n set, contrasting with the OD nature of non-empty Sigma^1_n sets.

Contribution

It introduces a modified invariant Jensen forcing to create models with specific definability properties of E_0 classes at arbitrary projective levels.

Findings

Existence of non-OD E_0 classes at any projective level n≥3.

Non-empty Sigma^1_n sets are necessarily constructible and OD.

The model distinguishes between E_0 classes and Sigma^1_n sets in terms of OD properties.

Abstract

Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable $\varSigma^1_n$ set of reals is necessarily constructible, hence contains only OD reals.

Equations

$${\text{spl}}_n(T)=\mathrm{lh}(u_0)+1+\mathrm{lh}(u_1)+1+\cdots +\mathrm{lh}(u_{n-1})+1+\mathrm{lh}(u_n),$$

$$T(\to i)=T{\upharpoonright }_{\text{stem}(T){}^{⌢}i}=\{x\in [T]:\text{linebreak}[0]x(h)=i\},\text{where }h=\mathrm{lh}(\text{stem}(T)),i=0,1.$$

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

$$\mathbf{P}=\{\sigma \cdot (T{\upharpoonright }_u):\text{linebreak}[0]u\in T\in Q\wedge \sigma \in 2^{<\omega }\}=\{\sigma \cdot (T(\to s)):\text{linebreak}[0]T\in Q\wedge s,\sigma \in 2^{<\omega }\}.$$

$$\cdots {\subseteq }_5{\mathbf{T}}_m^{\mathbf{\Phi }}(4){\subseteq }_4{\mathbf{T}}_m^{\mathbf{\Phi }}(3){\subseteq }_3{\mathbf{T}}_m^{\mathbf{\Phi }}(2){\subseteq }_2{\mathbf{T}}_m^{\mathbf{\Phi }}(1){\subseteq }_1{\mathbf{T}}_m^{\mathbf{\Phi }}(0)$$

$${\mathbf{U}}_m^{\mathbf{\Phi }}={\bigcap }_n{\mathbf{T}}_m^{\mathbf{\Phi }}(n)={\bigcap }_n{\bigcup }_{s\in 2^n}{\mathbf{T}}_m^{\mathbf{\Phi }}(s)$$

$${\mathbf{U}}_m^{\mathbf{\Phi }}(\to s)={\mathbf{U}}_m^{\mathbf{\Phi }}\cap {\mathbf{T}}_m^{\mathbf{\Phi }}(s)={\bigcap }_{n\ge \mathrm{lh}(s)}{\mathbf{T}}_m^{\mathbf{\Phi }}(n)(\to s),$$

$${\text{CB}}_{\mathbf{c}}=\{T\in \mathbf{ST}:\text{linebreak}[0]f_{\mathbf{c}}\upharpoonright [T]\text{ is a constant or a bijection}\}$$

$$[x{]}_{{\text{E}}_0}=\{y\in 2^{\omega }:\text{linebreak}[0]x{\text{E}}_{0}y\}=\{y\in 2^{\omega }:\text{linebreak}[0]\exists s\in 2^{<\omega }(y=s\cdot x)\}$$

$$\exists x_1\forall x_2\exists x_3\dots \exists (\forall )x_n\Psi \text{and}\forall x_1\exists x_2\forall x_3\dots \forall (\exists )x_n\Psi ,$$

Full text

1. 1 Introduction
2. 2 Connections to the Vitali equivalence relation
3. 3 Structure of the proof
4. 4 Silver trees
5. 5 Splitting Silver trees
6. 6 ST-forcings
7. 7 Continuous maps
8. 8 Generic extensions of ST-forcings
9. 9 Construction of extending ST-forcing
10. 10 Validation of the extension property
11. 11 The blocking sequence of ST-forcings
12. 12 CCC and some other forcing properties
13. 13 Generic model
14. 14 Definability of the set of generic reals
15. 15 Auxiliary forcing relation
16. 16 Invariance
17. 17 The final argument
18. Index

1 Introduction

Problems related to definability of mathematical objects, were one of focal points of the famous discussion on mathematical foundations in the beginning of XIX C. In particular, Baire, Borel, Hadamard, and Lebesgue, participants of the exchange of letters published in [4], in spite of essential disagreement between them on questions related to mathematical foundations, generally agreed that the proof of existence of an element in a given set, and a direct definition (or effective construction) of such an element — are different mathematical results, of which the second does not follow from the first. In particular, Lebesgue in his contribution to [4] pointed out the difficulties in the problem of effective choice, that is, choice of definable element in a definable non-empty set. 111 Ainsi je vois déjà une difficulté dans ceci “dans un $M^{\prime}$ déterminé je puis choisir un $m^{\prime}$ déterminé”, in the French original. Thus I already see a difficulty with the assertion that “in a determinate $M^{\prime}$ I can choose a determinate $m^{\prime}$”, in the translation.

Studies in modern set theory demonstrated that effective effective choice is not always possible. In particular, it is true in many well-known models (including the very first Cohen models), that the set $X={\hskip 0.43057pt{\mathbb{R}}\hskip 0.43057pt}\smallsetminus{\mathbf{L}}$ of all Goedel-nonconstructible reals is not empty, but contains no definable elements.

One may note that if the set $X$ is non-empty then it has to be rather large, that is, surely of cardinality $\mathfrak{c}$, if measurable then of full measure, etc.. Is there such an example among small, e. g. countable sets? This problem was discussed at Mathoverflow 222  A question about ordinal definable real numbers. Mathoverflow, March 09, 2010. http://mathoverflow.net/questions/17608.

and Foundations of mathematics (FOM) 333 Ali Enayat. Ordinal definable numbers. FOM Jul 23, 2010. http://cs.nyu.edu/pipermail/fom/2010-July/014944.html .

The problem was solved in [9] (to appear in [15]). Namely, let ${\mathbf{L}}[\langle a_{n}\rangle_{n<\omega}]$ be a $\hskip 0.86108pt{\boldsymbol{J}^{\omega}}$-generic extension of ${\mathbf{L}}$, where $\boldsymbol{J}^{\omega}$ is the countable power (with finite support) of Jensen’s minimal forcing $\boldsymbol{J}$ [8] 444 See also 28A [7] on this forcing. We acknowledge that the idea to use the countable power $\boldsymbol{J}^{\omega}$ of Jensen’s forcing $\boldsymbol{J}$ to define such a model belongs to Ali Enayat [2].. The key property of $\boldsymbol{J}$ is that it adds a nonconstructible generic $\varDelta^{1}_{3}$ real $a\in 2^{\omega}$, in fact $\{\hskip 0.43057pta\hskip 0.43057pt\}$ is a $\varPi^{1}_{2}$ singleton. Accordingly $\boldsymbol{J}^{\omega}$ adds a sequence $\langle a_{n}\rangle_{n<\omega}$ of $\hskip 0.86108pt{\boldsymbol{J}}$-generic reals real $a_{n}\in 2^{\omega}$ to ${\mathbf{L}}$, and as shown in [9, 15] there is no orher $\hskip 0.86108pt{\boldsymbol{J}}$-generic reals in ${\mathbf{L}}[\langle a_{n}\rangle_{n<\omega}]$ except for the reals $a_{n}$. Furthermore, since “being a $\hskip 0.86108pt{\boldsymbol{J}}$-generic real” is a $\varPi^{1}_{2}$ relation, it is true in ${\mathbf{L}}[\langle a_{n}\rangle_{n<\omega}]$ that $A=\{\hskip 0.43057pt{a_{n}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptn<\omega\hskip 0.43057pt\}$ is a countable (infinite) lightface $\varPi^{1}_{2}$ set without OD (ordinal-definable) elements.

Using an uncountable product of forcing notions similar to $\boldsymbol{J}^{\omega}$, we defined in [11] a model in which there is a “planar” $\varPi^{1}_{2}$ set with countable vertical cross-sections, which cannot be uniformized by any real-ordinal definable (ROD) set.

For a more detailed analysis of the problem, note that the elements $a_{n}$ of the set $A$, adjoined by the forcing $\boldsymbol{J}^{\omega}$, are connected to each other only by the common property of their $\hskip 0.86108pt{\boldsymbol{J}}$-genericity. Does there exist a similar countable set with a more definite mathematical structure?

This question was answered in [10] by a model in which there is an equivalence class of the equivalence relation $\mathrel{{\text{\sf E}}_{0}}$ 555 Recall that $\mathrel{{\text{\sf E}}_{0}}$ is defined on the Cantor space $2^{\omega}$ so that $x\mathrel{{\text{\sf E}}_{0}}y$ iff the set $\{\hskip 0.43057pt{n\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptx(n)\neq y(n)\hskip 0.43057pt\}$ is finite. $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-equivalence classes are countable sets in $2^{\omega}$, of course. (a $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class, for brevity), which is a (countable) lightface $\varPi^{1}_{2}$ set in $2^{\omega}$, and does not contain OD elements. This model makes use of a forcing notion $\boldsymbol{J}^{\text{inv}}$, similar to Jensen’s forcing $\boldsymbol{J}$, but different from $\boldsymbol{J}$. In particular, it consists of Silver trees (rather than perfect trees of general form, as $\boldsymbol{J}$ does) and is invariant under finite transformations. Thus it can be called an invariant Jensen forcing. Due to the invariance, $\boldsymbol{J}^{\text{inv}}$ adjoins a $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-equivalence class of $\hskip 0.86108pt{\boldsymbol{J}^{\text{inv}}}$-generic reals rather than a single real, and this class turns out to be a $\varPi^{1}_{2}$ set without OD elements. A forcing similar to $\boldsymbol{J}^{\text{inv}}$ was also used in [3] to define a model with a $\varPi^{1}_{2}$ Groszek – Laver pair of $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-classes.

Our main theorem extends this research line.

Theorem 1.1**.**

Let $\mathbb{n}\geq 3$. There is a generic extension ${\mathbf{L}}[a]$ of ${\mathbf{L}}$, by a real $a\in 2^{\omega}$, such that the following is true in {\mathbf{L}}[a]$$:

1. (i)

$a\notin\text{\rm OD}$* and $a$ is minimal over ${\mathbf{L}}$ — hence each OD real belongs to ${\mathbf{L}}\,;$* 2. (ii)

the $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class $[a]_{\mathrel{{\text{\sf E}}_{0}}}$ is a (countable) lightface $\varPi^{1}_{\mathbb{n}}$ set — which by (i) does not contain OD elements*;*** 3. (iii)

every countable $\varSigma^{1}_{\mathbb{n}}$ set belongs to ${\mathbf{L}}$, hence consists of OD elements.

Thus we have a model, in which, for a given $\mathbb{n}\geq 3$, there is a $\varPi^{1}_{\mathbb{n}}$, hence, OD $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class (therefore, a countable set) in $2^{\omega}$ containing only non-OD elements, and in the same time every countable $\varSigma^{1}_{\mathbb{n}}$ set belongs to ${\mathbf{L}}$, hence consists of OD elements. The abovementioned result of [10] corresponds to $\mathbb{n}=2$ in this theorem since countable $\varSigma^{1}_{2}$ sets belong to ${\mathbf{L}}$ and consist of OD elements.

2 Connections to the Vitali equivalence relation

The relation $\mathrel{{\text{\sf E}}_{0}}$ is tightly connected with the Vitali equivalence relation $\mathrel{\text{\sf VIT}}={\hskip 0.43057pt{\mathbb{R}}\hskip 0.43057pt}/{\hskip 0.43057pt{\mathbb{Q}}\hskip 0.43057pt}$ on the true real line ${\hskip 0.43057pt{\mathbb{R}}\hskip 0.43057pt}$. 666 $\mathrel{\text{\sf VIT}}$ is defined on ${\hskip 0.43057pt{\mathbb{R}}\hskip 0.43057pt}$ so that $x\mathrel{\text{\sf VIT}}y$ iff the set $x-y$ is rational. In particular, there is a lightface $\varDelta^{1}_{1}$ (in fact of a very low class $\varDelta^{0}_{n}$) injection $\vartheta:R\to 2^{\omega}$ which reduces $\mathrel{\text{\sf VIT}}$ to $\mathrel{{\text{\sf E}}_{0}}$ in the sense that the equivalence ${x\mathrel{\text{\sf VIT}}y}\mathbin{\,\Longleftrightarrow\,}{\vartheta(x)\mathrel{{\text{\sf E}}_{0}}\vartheta(y)}$ holds for all $x,y\in{\hskip 0.43057pt{\mathbb{R}}\hskip 0.43057pt}$. (See Mycielski and Osofsky [17] for an explicit construction of $\vartheta$, along with an inverse reduction of $\mathrel{{\text{\sf E}}_{0}}$ to $\mathrel{\text{\sf VIT}}$.) It follows that Theorem 1.1 is true with (ii) for $[a]_{\mathrel{\text{\sf VIT}}}$ as well, since if $C\subseteq 2^{\omega}$ is a $\varPi^{1}_{\mathbb{n}}$ $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class not containing OD elements then the preimage $C^{\prime}=\{\hskip 0.43057pt{x\in{\hskip 0.43057pt{\mathbb{R}}\hskip 0.43057pt}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\vartheta(x)\in C\hskip 0.43057pt\}$ is a $\varPi^{1}_{\mathbb{n}}$ Vitali class not containing OD elements.

The interest to Vitali classes in this context is inspired by the observation that they can be viewed as the most elementary countable sets in $\mathbb{R}$ which do not allow immediate effective choice of an element. Indeed if a set $X\subseteq\mathbb{R}$ contains at least one isolated, or even one-sided isolated point, then one of such points can be chosen effectively. However any non-empty set $X\subseteq\mathbb{R}$ without one-sided isolated points is just an everywhere dense set (not counting close segments of the complementary set). Yet the Vitali classes, that is, shifts of the rationals $\mathbb{Q}$, are exactly the most simple and typical countable dense sets in $\mathbb{R}$.

Historically, the Vitali relation and its equivalence classes have deep roots in descriptive set theory. For instance Sierpinski [18, c. 147] and Luzin [16, Section 64] observed that the quotient set $\mathbb{R}/\mathbb{Q}$ of all Vitali classes has the property that () it cannot be mapped into $\mathbb{R}$ by an injective Borel map. On the other hand, as established in [14], in models of ZF (without the axiom of choice) the Hartogs number of the set $\mathbb{R}/\mathbb{Q}$ (the least cardinal which cannot be injectively mapped into $\mathbb{R}/\mathbb{Q}$) can be greater than the continuum. The relations $\mathrel{{\text{\sf E}}_{0}}$ and $\mathrel{\text{\sf VIT}}$ play a key role in modern studies of Borel equivalence relations, being the least ones, in the sense of the Borel reducibility [5], among those satisfying ().

3 Structure of the proof

The proof of Theorem 1.1 is arganized as follows. Basic notions, related to Silver trees in the set $2^{<\omega}$ of all finite dyadic strings, are introduced in sections 4–7. Every set ${\mathbf{P}}$ of Silver trees $T$, closed under restriction to a given string $s\in T$, and under the natural action $s\boldsymbol{\cdot}T$ by $s\in 2^{<\omega}$, is considered as a forcing by Silver trees, a ST-forcing, in brief. Every ST-forcing adjoins a $\hskip 0.86108pt{\mathbf{P}}$-generic real $a\in 2^{\omega}.$

Arguing in the constructible universe ${\mathbf{L}}$, we define a forcing notion to prove Theorem 1.1 in Section 11 in the form ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}=\bigcup_{\alpha<\omega_{1}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$. The summands ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ are countable ST-forcings defined by induction. Any $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic extension of ${\mathbf{L}}$ is a model for Theorem 1.1. The inductive construction of ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ involves two key ideas.

The first idea, essentially by Jensen [8], is to make every level ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ of the construction generic in some sense over the union of lower levels ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}$, $\xi<\alpha$. This is based on a construction developed in sections 8 – 10, which includes the technique of fusion of Silver trees. A special aspect of this construction, elaborated in [10], guarantees that ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is invariant under the group of transformations ($2^{<\omega}$ with the componentwise addition mod 2), which induces the equivalence relation $\mathrel{{\text{\sf E}}_{0}}$. This invariance implies that ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ (unless Jensen’s original forcing) adjoins a $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class of generic reals rather than a single such real as in [8]. And overall, the successive genericity of the levels ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ implies that the three sets are equal in any $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic extension of ${\mathbf{L}}$:

the $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class $[a]_{\mathrel{{\text{\sf E}}_{0}}}$ of the principal generic real $a\in 2^{\omega}$,

the intersection $\bigcap_{\alpha<\omega_{1}}\bigcup_{T\in{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}_{\alpha}}$,

and the set of all $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic reals over ${\mathbf{L}}$.

This equality, which leads to $[a]_{\mathrel{{\text{\sf E}}_{0}}}$ being $\varPi^{1}_{\mathbb{n}}$, is established in sections 12 – 14.

The second idea goes back to old papers [6, 13]. In ${\mathbf{L}}$, let $\overline{\mathbf{STF}}$ be the set of all countable sequences $\overline{\mathbf{P}}=\langle\mathbf{P}_{\xi}\rangle_{\xi<\alpha}$ ($\alpha<\omega_{1}$) compatible with the first genericity idea. Then a whole sequence $\langle\mathbf{P}_{\alpha}\rangle_{\alpha<\omega_{1}}$ can be interpreted as a maximal branch in $\overline{\mathbf{STF}}$. It happens that if this branch is generic, in some sense precisely defined in Section 11 ((ii) of Theorem 11.4), with respect to all ${\mathbf{\Sigma}}^{1}_{\mathbb{n}-1}$ subsets of $\overline{\mathbf{STF}}$, then the ensuing forcing notion ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}=\bigcup_{\alpha<\omega_{1}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ inherits some basic forcing properties of the whole Silver forcing, up to a certain level of projective hierarchy. This includes, in particular, the invariance of the forcing relation with respect to some natural transformations of Silver trees, leading eventually to the proof of (iii) of Theorem 1.1 in sections 15 – 17. But before that the actual construction of ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}=\bigcup_{\alpha<\omega_{1}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ involving both mentioned ideas is given in Section 11.

4 Silver trees

Let $2^{<\omega}$ be the set of all dyadic strings (finite sequences) of numbers $0,1$ — including the empty string $\Lambda$. If $t\in 2^{<\omega}$ and $i=0,1$, then $t{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i$ is the extension of $t$ by $i$ as the rightmost term. If $s,t\in 2^{<\omega}$ then $s\subseteq t$ means that the string $t$ extends $s$, while $s\subset t$ means a proper extension. The length of a string $s$ is denoted by $\mathop{\tt lh}(s)$, and we let $2^{n}=\{\hskip 0.43057pt{s\in 2^{<\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\mathop{\tt lh}(s)=n\hskip 0.43057pt\}$ (strings of length $n$).

As usual, $2^{\omega}$ is the set of all functions $f:\omega\to 2=\{\hskip 0.43057pt0,1\hskip 0.43057pt\}$, — the Cantor space. Any string $s\in 2^{<\omega}$ acts on $2^{\omega}$ so that $(s\boldsymbol{\cdot}x)(k)=x(k)+s(k)\pmod{2}$ whenever $k<\mathop{\tt lh}(s)$, and $(s\boldsymbol{\cdot}x)(k)=x(k)$ otherwise. If $X\subseteq 2^{\omega}$ and $s\in 2^{<\omega}$ then let $s\boldsymbol{\cdot}X=\{\hskip 0.43057pt{s\boldsymbol{\cdot}x\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptx\in X\hskip 0.43057pt\}$.

Similarly, if $s\in 2^{m},$ $t\in 2^{n},$ $m\leq n$, then define a string $s\boldsymbol{\cdot}t\in 2^{n}$ by $(s\boldsymbol{\cdot}t)(k)=t(k)+s(k)\pmod{2}$ whenever $k<m$, and $(s\boldsymbol{\cdot}t)(k)=t(k)$ whenever $m\leq k<n$. If $m>n$, then let $s\boldsymbol{\cdot}t=(s{\hskip 0.43057pt\restriction\hskip 1.29167pt}n)\boldsymbol{\cdot}t$. In both cases, $\mathop{\tt lh}(s\boldsymbol{\cdot}t)=\mathop{\tt lh}(t)$. If $T\subseteq 2^{<\omega}$ then let $s\boldsymbol{\cdot}T=\{\hskip 0.43057pt{s\boldsymbol{\cdot}t\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptt\in T\hskip 0.43057pt\}$.

Definition 4.1**.**

A set $T\subseteq 2^{<\omega}$ is a Silver tree, in symbol $T\in\mathbf{ST}$, whenever there exists an infinite sequence of strings $u_{k}=u_{k}(T)\in 2^{<\omega}$ such that $T$ consists of all strings of the form $s=u_{0}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i_{0}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}u_{1}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i_{1}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}u_{2}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i_{2}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}\dots\mathbin{\hskip 0.86108pt{}^{\smallfrown}}u_{n}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i_{n}$, and their substrings (including $\Lambda$), where $n<\omega$ and $i_{k}=0,1$.

In this case we let $\text{\tt stem}(T)=u_{0}$ (the stem of $T$), and define a closed set $[T]\subseteq 2^{\omega}$ of all branches of $T$, i. e. all sequences $a=u_{0}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i_{0}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}u_{1}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i_{1}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}u_{2}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i_{2}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}\dots\in 2^{\omega},$ where $i_{k}=0,1$, ${\forall\,}k$. Let

[TABLE]

in particular, $\text{\tt spl}_{0}(T)=\mathop{\tt lh}(u_{0})$, so that $\text{\tt spl}(T)=\{\hskip 0.43057pt{\text{\tt spl}_{n}(T)\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptn<\omega\hskip 0.43057pt\}\subseteq\omega$ is the set of all splitting levels of $T\in\mathbf{ST}$; $\text{\tt spl}(T)$ is infinite. If $u\in T\in\mathbf{ST}$, then define the restricted tree $T{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}=\{\hskip 0.43057pt{t\in T\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptu\subseteq t\lor t\subseteq u\hskip 0.43057pt\}$; $T{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}\in\mathbf{ST}$. ∎

Example 4.2**.**

If $s\in 2^{<\omega}$ then the tree $B[s]=\{\hskip 0.43057pt{t\in 2^{<\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\subseteq t\lor t\subset s\hskip 0.43057pt\}$ belongs to $\mathbf{ST}$, $\text{\tt stem}(B[s])=u_{0}(B[s])=s$, $\text{\tt spl}_{n}(B[s])=\mathop{\tt lh}(s)+n$, and if $k\geq 1$ then $u_{k}(B[s])=\Lambda$. In particular $B[\Lambda]=2^{<\omega}$. ∎

The corresponding sets $[T[s]]=\{\hskip 0.43057pt{x\in\omega^{\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\subset x\hskip 0.43057pt\}$ are clopen in $2^{\omega}$.

Lemma 4.3**.**

Let $T\in\mathbf{ST}$. Then

1. (i)

if a set $X\subseteq 2^{\omega}$ is open and $X\cap[T]\neq\varnothing$, then there is a string $s\in T$ such that ${T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507pts}\subseteq X\;;$ 2. (ii)

if $h\in\text{\tt spl}(T)$ and $u,v\in 2^{h}\cap T$ then ${T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptv}=(u\boldsymbol{\cdot}v)\boldsymbol{\cdot}{T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}$ and $(u\boldsymbol{\cdot}v)\boldsymbol{\cdot}T=T$.

Proof**.**

(i) If $a\in X\cap[T]$ then $\{\hskip 0.43057pta\hskip 0.43057pt\}=\bigcap_{n}{T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507pta{\hskip 0.3014pt\restriction\hskip 0.90417pt}n}$, and hence ${T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507pta{\hskip 0.3014pt\restriction\hskip 0.90417pt}n}\subseteq X$ for some $n$ by the compactness. Let $s=a{\hskip 0.43057pt\restriction\hskip 1.29167pt}n$. ∎

5 Splitting Silver trees

The simple splitting of a tree $T\in\mathbf{ST}$ consists of the subtrees

[TABLE]

Then we have $T{({\to\hskip 0.86108pt}i)}\in\mathbf{ST}$, and $u_{0}(T{({\to\hskip 0.86108pt}i)})=u_{0}(T){\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}\langle i\rangle{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}u_{1}(T)$, and then $u_{k}(T{({\to\hskip 0.86108pt}i)})=u_{k+1}(T)$ whenever $k\geq 1$, and $\text{\tt spl}(T{({\to\hskip 0.86108pt}i)})=\text{\tt spl}(T)\smallsetminus\{\hskip 0.43057pt\text{\tt spl}_{0}(T)\hskip 0.43057pt\}$.

The splitting can be iterated. Namely if $s\in 2^{n}$ then define

[TABLE]

Separately let $T{({\to\hskip 0.86108pt}\Lambda)}=T$, for the empty string $\Lambda$.

Lemma 5.1**.**

Let $T\in\mathbf{ST}$, $n<\omega$, and $h=\text{\tt spl}_{n}(T)$. Then

1. (i)

if $s\in 2^{n}$ then $T{({\to\hskip 0.86108pt}s)}\in\mathbf{ST}$, $\mathop{\tt lh}(\text{\tt stem}(T{({\to\hskip 0.86108pt}s)}))=h$, and there is a string $u[s]\in 2^{h}\cap T$ such that $T{({\to\hskip 0.86108pt}s)}={T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu[s]}\;;$ 2. (ii)

conversely if $u\in 2^{h}\cap T$, then there is a string $s[u]\in 2^{n}$ such that ${T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}=T{({\to\hskip 0.86108pt}s[u])}\;;$ 3. (iii)

therefore, $\{\hskip 0.43057pt{{T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptu\in T\hskip 0.43057pt\}=\{\hskip 0.43057pt{T{({\to\hskip 0.86108pt}s)}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\in 2^{<\omega}\hskip 0.43057pt\}\;.$

Proof**.**

(i) $u[s]=u_{0}(T){\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}\langle s(0)\rangle{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}\dots\mathbin{\hskip 0.86108pt{}^{\smallfrown}}u_{n-1}(T){\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}\langle s(n-1)\rangle{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}u_{n}(T)$.

(ii) Define $s=s[u]\in 2^{n}$ by $s(k)=u(\text{\tt spl}_{k}(T))$ for all $k<n$.

(iii) Let $u\in T$. Then $\text{\tt spl}_{n-1}(T)<\mathop{\tt lh}(u)\leq\text{\tt spl}_{n}(T)$ for some $n$. Now, by Definition 4.1, there exists a (unique) string $v\in 2^{h}\cap T$, where $h=\text{\tt spl}_{n}(T)$, such that ${T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}={T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptv}$. It remains to refer to (ii). ∎

If $T,S\in\mathbf{ST}$ and $n\in\omega$ then define $S\mathrel{{\subseteq}_{n}}T$

(the tree $S$ $\hskip 0.86108pt{n}$-refines $T$), whenever $S\subseteq T$ and $\text{\tt spl}_{k}(T)=\text{\tt spl}_{k}(S)$ for all $k<n$. In particular $S\mathrel{{\subseteq}_{0}}T$ is equivalent to just $S\subseteq T$. By definition if $S\mathrel{{\subseteq}_{n+1}}T$ then $S\mathrel{{\subseteq}_{n}}T$ (and $S\subseteq T$).

Lemma 5.2**.**

If $T\in\mathbf{ST}$, $n<\omega$, $s_{0}\in 2^{n}$, and $U\in\mathbf{ST}$, $U\subseteq T{({\to\hskip 0.86108pt}s_{0})}$, then there is a unique tree $T^{\prime}\in\mathbf{ST}$ such that $T^{\prime}\mathrel{{\subseteq}_{n}}T$ and $T^{\prime}{({\to\hskip 0.86108pt}s_{0})}=U$.

Proof**.**

Let $h=\text{\tt spl}_{n}(T)$. Pick a string $u_{0}=u[s_{0}]\in 2^{h}$ by Lemma 5.1(i). Following Lemma 4.3(ii), define $T^{\prime}$ so that $T^{\prime}\cap 2^{h}=T\cap 2^{h}$, and if $u\in T\cap 2^{h}$ then ${T^{\prime}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}=(u\cdot u_{0})\cdot U$. In particular ${T^{\prime}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu_{0}}=U$. ∎

Lemma 5.3**.**

Let $\dots\mathrel{{\subseteq}_{5}}T_{4}\mathrel{{\subseteq}_{4}}T_{3}\mathrel{{\subseteq}_{3}}T_{2}\mathrel{{\subseteq}_{2}}T_{1}\mathrel{{\subseteq}_{1}}T_{0}$ be an infinite decreasing sequence of trees $\mathbf{ST}$. Then

1. (i)

$T=\bigcap_{n}T_{n}\in\mathbf{ST}\;;$** 2. (ii)

if $n<\omega$ and $s\in 2^{n+1}$ then $T{({\to\hskip 0.86108pt}s)}=T\cap T_{n}{({\to\hskip 0.86108pt}s)}=\bigcap_{m\geq n}T_{m}{({\to\hskip 0.86108pt}s)}$.

Proof**.**

Note that $\text{\tt spl}(T)=\{\hskip 0.43057pt{\text{\tt spl}_{n}(T_{n})\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptn<\omega\hskip 0.43057pt\}$; both claims easily follow. ∎

The next application of this lemma and Lemma 5.2 is well known:

Corollary 5.4**.**

Let $T\in\mathbf{ST}$. If a set $X\subseteq[T]$ has the Baire property inside $[T]$, but is not meager inside $[T]$, then there is a tree $S\in\mathbf{ST}$ such that $[S]\subseteq X$. If $f:[T]\to 2^{\omega}$ is a continuous map then there exists a tree $S\in\mathbf{ST}$ such that $S\subseteq T$ and $f{\hskip 0.43057pt\restriction\hskip 1.29167pt}[S]$ is a bijection or a constant.∎

6 ST-forcings

Definition 6.1**.**

Any set $\mathbf{P}\subseteq\mathbf{ST}$ satisfying

1. (A)

if $u\in T\in{\mathbf{P}}$ then ${T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}\in{\mathbf{P}}$, and 2. (B)

if $T\in{\mathbf{P}}$ and $\sigma\in 2^{<\omega}$ then $\sigma\boldsymbol{\cdot}T\in{\mathbf{P}}$,

is called a forcing by Silver trees, ST-forcing in brief. ∎

Remark 6.2**.**

Any ST-forcing ${\mathbf{P}}$ can be considered as a forcing notion ordered so that if $T\subseteq T^{\prime}$, then $T$ is a stronger condition. The forcing ${\mathbf{P}}$ adjoins a real $x\in 2^{\omega}$. More exactly if a set $G\subseteq{\mathbf{P}}$ is $\hskip 0.86108pt{\mathbf{P}}$-generic over a given model or set universe $M$ (and ${\mathbf{P}}\in M$ is assumed) then the intersection $\bigcap_{T\in G}[T]$ contains a unique real $a=\mathbf{a}[G]\in 2^{\omega}$, and this real satisfies $M[G]=M[\mathbf{a}[G]]$ and $G=\{\hskip 0.43057pt{T\in{\mathbf{P}}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\mathbf{a}[G]\in[T]\hskip 0.43057pt\}$. Reals $\mathbf{a}[G]$ of this kind are called $\hskip 0.86108pt{\mathbf{P}}$-generic. ∎

Example 6.3**.**

The following sets are ST-forcings: the set $\mathbf{P}_{\text{\rm coh}}=\{\hskip 0.43057pt{T[s]\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\in 2^{<\omega}\hskip 0.43057pt\}$ of all trees in 4.2 — the Cohen forcing, and the set $\mathbf{ST}$ of all Silver trees — the Silver forcing itself. ∎

Lemma 6.4**.**

If $\varnothing\neq Q\subseteq\mathbf{ST}$ then the following set is a ST-forcing$:$

[TABLE]

Proof**.**

To prove 6.1(A), let $T\in Q$ and $v\in S=\sigma\boldsymbol{\cdot}({T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu})$. Then $w=\sigma\boldsymbol{\cdot}v\in{T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}$ and $v=\sigma\boldsymbol{\cdot}w$. It follows that ${S}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptv}=\sigma\boldsymbol{\cdot}({{T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptw})$, where ${{T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptw}={T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}$ or $={T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptw}$, whenever accordingly $w\subseteq u$ or $u\subset w$. The second equality of the lemma follows from Lemma 5.1(iii). ∎

Definition 6.5** (collages).**

If ${\mathbf{P}}\subseteq\mathbf{ST}$, $T\in\mathbf{ST}$, $n<\omega$, and all split trees $T{({\to\hskip 0.86108pt}s)}$, $s\in 2^{n}$, belong to ${\mathbf{P}}$ then $T$ is an $\hskip 0.86108pt{n}$-collage over ${\mathbf{P}}$. The set of all $\hskip 0.86108pt{n}$-collages over ${\mathbf{P}}$ is $\text{Colg}_{n}(\mathbf{P})$. Then ${\mathbf{P}}=\text{Colg}_{0}(\mathbf{P})\subseteq\text{Colg}_{n}(\mathbf{P})\subseteq\text{Colg}_{n+1}(\mathbf{P})$. ∎

Lemma 6.6**.**

Let ${\mathbf{P}}\subseteq\mathbf{ST}$ be a ST-forcing, and $n<\omega$. Then$:$

1. (i)

if $T\in{\mathbf{P}}$ and $s\in 2^{<\omega}$ then $T{({\to\hskip 0.86108pt}s)}\in{\mathbf{P}}\;;$ 2. (ii)

if $T\in\mathbf{ST}$ and $s_{0}\in 2^{n}$ then $T{({\to\hskip 0.86108pt}s_{0})}\in{\mathbf{P}}$ is equivalent to $T\in\text{Colg}_{n}(\mathbf{P})$;** 3. (iii)

if $U\in\text{Colg}_{n}(\mathbf{P})$, $s_{0}\in 2^{n}$, $S\in{\mathbf{P}}$, and $S\subseteq U{({\to\hskip 0.86108pt}s_{0})}$ then there is a tree $V\in\text{Colg}_{n}(\mathbf{P})$ such that $V\mathrel{{\subseteq}_{n}}U$ and $V{({\to\hskip 0.86108pt}s_{0})}=S\;;$ 4. (iv)

if $U\in\text{Colg}_{n}(\mathbf{P})$ and $D\subseteq{\mathbf{P}}$ is open dense in ${\mathbf{P}}$ then there is a tree $V\in\text{Colg}_{n}(\mathbf{P})$ such that $V\mathrel{{\subseteq}_{n}}U$ and $V{({\to\hskip 0.86108pt}s)}\in D$ for all $s\in 2^{n}$.

A set $D\subseteq{\mathbf{P}}$ is dense in ${\mathbf{P}}$ if for any $S\in{\mathbf{P}}$ there is a tree $T\in D$, $T\subseteq S$, and open dense, if in addition $S\in D$ holds whenever $S\in{\mathbf{P}}$, $T\in D$, $S\subseteq T$.

Proof**.**

To prove (i) make use of 6.1(A) and Lemma 5.1(i).

(ii) If $T\in\text{Colg}_{n}(\mathbf{P})$ then by definition $T{({\to\hskip 0.86108pt}s_{0})}\in{\mathbf{P}}$. To prove the converse let $h=\text{\tt spl}_{n}(T)$ and let a string $u[s]\in 2^{h}\cap T$ satisfy $T{({\to\hskip 0.86108pt}s)}={T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu[s]}$ for all $s\in 2^{n}$ by lemma 5.1(i). If $T{({\to\hskip 0.86108pt}s_{0})}\in{\mathbf{P}}$ and $s\in 2^{n}$ then $T{({\to\hskip 0.86108pt}s)}={T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu[s]}=(u[s]\boldsymbol{\cdot}u[s_{0}])\boldsymbol{\cdot}{T}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu[s_{0}]}=(u[s]\boldsymbol{\cdot}u[s_{0}])\boldsymbol{\cdot}T{({\to\hskip 0.86108pt}s_{0})}$ by Lemma 4.3(ii). Therefore $T{({\to\hskip 0.86108pt}s)}\in{\mathbf{P}}$ by 6.1(B). And finally $T\in\text{Colg}_{n}(\mathbf{P})$.

(iii) By Lemma 5.2, there is a tree $V\in\mathbf{ST}$, satisfying $V\mathrel{{\subseteq}_{n}}U$ and $V{({\to\hskip 0.86108pt}s_{0})}=S$. However $V\in\text{Colg}_{n}(\mathbf{P})$ by (ii).

To prove (iv) apply (iii) $2^{n}$ times (for all $s\in 2^{n}$). ∎

7 Continuous maps

Let ${\mathbf{P}}$ be a fixed ST-forcing in this section.

Regularity. We study the behaviour of continuous maps on sets of the form $[T]$, $T\in{\mathbf{P}}$. The next definition highlights the case when a given continuous $f:2^{\omega}\to 2^{\omega}$ is forced to be not equal to a map of the form $x\mapsto\sigma\boldsymbol{\cdot}x$, $\sigma\in 2^{<\omega}.$

Definition 7.1**.**

Let $T\in{\mathbf{P}}$. A continuous $f:2^{\omega}\to 2^{\omega}$ is regular on $T$ inside ${\mathbf{P}}$, if there is no tree $T^{\prime}\in{\mathbf{P}}$ and string $\sigma\in 2^{<\omega}$ such that $T^{\prime}\subseteq T$ and $\sigma\boldsymbol{\cdot}f(x)=x$ (equivalently, $f(x)=\sigma\boldsymbol{\cdot}x$) for all $x\in[T^{\prime}]$. ∎

Lemma 7.2**.**

Let $S,T\in{\mathbf{P}}$, $f:2^{\omega}\to 2^{\omega}$ be continuous, and $\sigma\in 2^{<\omega}$. Then$:$

1. (i)

there are trees $S^{\prime},T^{\prime}\in{\mathbf{P}}$ such that $S^{\prime}\subseteq S$, $T^{\prime}\subseteq T$, $[T^{\prime}]\cap(\sigma\boldsymbol{\cdot}{f}\text{\rm\hskip 0.86108pt''}{[}S^{\prime}])=\varnothing\;;$ 2. (ii)

if $\tau\in 2^{<\omega}$, $T=\tau\boldsymbol{\cdot}S$, and $f$ is regular on $S$ inside ${\mathbf{P}}$, then there exist trees $S^{\prime},T^{\prime}\in{\mathbf{P}}$ such that $S^{\prime}\subseteq S$, $T^{\prime}\subseteq T$, $T^{\prime}=\tau\boldsymbol{\cdot}S^{\prime}$, and $[T^{\prime}]\cap(\sigma\boldsymbol{\cdot}{f}\text{\rm\hskip 0.86108pt''}{[}S^{\prime}])=\varnothing\;.$

Proof**.**

(i) Let $x_{0}\in[S]$ and let $y_{0}\in[T]$ be different from $\sigma\boldsymbol{\cdot}f(x_{0})$. As $f$ is continuius, there is $m\geq\mathop{\tt lh}(\sigma)$, such that $\sigma\boldsymbol{\cdot}(f(x){\hskip 0.43057pt\restriction\hskip 1.29167pt}m)\neq y_{0}{\hskip 0.43057pt\restriction\hskip 1.29167pt}m$ whenever $x\in[S]$ and $x{\hskip 0.43057pt\restriction\hskip 1.29167pt}m=x_{0}{\hskip 0.43057pt\restriction\hskip 1.29167pt}m$. By 6.1(A), there are trees $S^{\prime},T^{\prime}\in{\mathbf{P}}$ such that $S^{\prime}\subseteq S$, $T^{\prime}\subseteq T$, $x{\hskip 0.43057pt\restriction\hskip 1.29167pt}m=x_{0}{\hskip 0.43057pt\restriction\hskip 1.29167pt}m$ for all $x\in[S^{\prime}]$, and $y{\hskip 0.43057pt\restriction\hskip 1.29167pt}m=y_{0}{\hskip 0.43057pt\restriction\hskip 1.29167pt}m$ for all $y\in[T^{\prime}]$.

(ii) Assume that $\mathop{\tt lh}(\sigma)=\mathop{\tt lh}(\tau)$ (otherwise the shorter string can be extended by zeros). The set $X=\{\hskip 0.43057pt{x\in[S]\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\sigma\boldsymbol{\cdot}f(x)\neq\tau\boldsymbol{\cdot}x\hskip 0.43057pt\}$ is open in $[S]$ and non-empty, by the regularity. Let $x_{0}\in X$. There is a number $m\geq\mathop{\tt lh}(\sigma)=\mathop{\tt lh}(\tau)$ satisfying $\tau\boldsymbol{\cdot}(x_{0}{\hskip 0.43057pt\restriction\hskip 1.29167pt}m)\neq\sigma\boldsymbol{\cdot}(f(x_{0}){\hskip 0.43057pt\restriction\hskip 1.29167pt}m)$. By 6.1(A) there is a tree $S^{\prime}\in{\mathbf{P}}$ satisfying $x_{0}\in[S^{\prime}]$ and $x{\hskip 0.43057pt\restriction\hskip 1.29167pt}m=x_{0}{\hskip 0.43057pt\restriction\hskip 1.29167pt}m$, $f(x){\hskip 0.43057pt\restriction\hskip 1.29167pt}m=f(x_{0}){\hskip 0.43057pt\restriction\hskip 1.29167pt}m$ for all $x\in[S^{\prime}]$. Put $T^{\prime}=\tau\boldsymbol{\cdot}S^{\prime}$. ∎

Coding continuous maps. If $f:2^{\omega}\to 2^{\omega}$ is continuous, $k<\omega$ and $i=0,1$ then the set ${\mathbf{D}}_{k}^{i}(f)=\{\hskip 0.43057pt{x\in 2^{\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptf(x)(k)=i\hskip 0.43057pt\}$, and ${\mathbf{D}}_{k}^{0}(f)\cap{\mathbf{D}}_{k}^{1}(f)=\varnothing$, ${\mathbf{D}}_{k}^{0}(f)\cup{\mathbf{D}}_{k}^{1}(f)=2^{\omega}$. Let ${\mathbf{C}}_{k}^{i}(f)$ be the set of all $\hskip 0.86108pt{\subseteq}$-minimal strings $s\in 2^{<\omega}$ such that $B[s]\subseteq{\mathbf{D}}_{k}^{i}(f)$. Then ${\mathbf{C}}_{k}^{0}(f),{\mathbf{C}}_{k}^{1}(f)\subseteq 2^{<\omega}$ are finite disjoint irreducible antichains 777 An antichain $A\subseteq 2^{<\omega}$ is irreducible, if it does not contain a pair of the form $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}0,s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}1$. in $2^{<\omega}$, and ${\mathbf{D}}_{k}^{i}(f)=\bigcup_{s\in{\mathbf{C}}_{k}^{i}(f)}B[s]$, while the union ${\mathbf{C}}_{k}^{0}(f)\cup{\mathbf{C}}_{k}^{1}(f)$ is a maximal antichain in $2^{<\omega}.$ Let ${\text{code}}(f)=\langle{\mathbf{C}}_{k}^{i}(f)\rangle_{k<\omega,i=0,1}$ — the code of $f$.

The other way around, if ${\mathbf{c}}=\langle{\mathbf{C}}_{k}^{i}\rangle_{k<\omega,i=0,1}$ is a family of finite irreducible antichains ${\mathbf{C}}_{k}^{i}\subseteq 2^{<\omega}$, and if $k<\omega$ then ${\mathbf{C}}_{k}^{0}\cap{\mathbf{C}}_{k}^{1}=\varnothing$ while ${\mathbf{C}}_{k}^{0}\cup{\mathbf{C}}_{k}^{1}$ is a maximal antichain in $2^{<\omega}$, then ${\mathbf{c}}$ is called a code of continuous function. In this case, the related continuous function $f:2^{\omega}\to 2^{\omega}$ is denoted by $f_{\mathbf{c}}$, that is, $f_{\mathbf{c}}(x)(k)=i$ whenever $x\in\bigcup_{s\in{\mathbf{C}}_{k}^{i}}B[s]$.

Let CCF denote the set of all codes of continuous functions.

8 Generic extensions of ST-forcings

The forcing notion to prove Theorem 1.1 will be defined in the form of an $\hskip 0.86108pt{\omega_{1}}$-union of its countable parts — levels. The next definition presents requirements which will govern the interactions between the levels.

Definition 8.1**.**

Let ${\mathfrak{M}}$ be any set and $\mathbf{P}$ be a ST-forcing. We say that another ST-forcing $\mathbf{Q}$ is an $\hskip 0.86108pt{{\mathfrak{M}}}$-extension of $\mathbf{P}$, in symbol $\mathbf{P}\mathrel{\sqsubset_{{\mathfrak{M}}}}\mathbf{Q}$, if the following holds:

1. (A)

$\mathbf{Q}$ is dense in $\mathbf{Q}\cup\mathbf{P}$; 2. (B)

if a set $D\in{\mathfrak{M}}$, $D\subseteq\mathbf{P}$ is pre-dense in $\mathbf{P}$ and $U\in\mathbf{Q}$ then $U\subseteq^{\text{\tt fin}}\bigcup D$, that is, there is a finite set $D^{\prime}\subseteq D$ such that $U\subseteq\bigcup D^{\prime}$; 3. (C)

if $T_{0}\in\mathbf{P}$, $f:2^{\omega}\to 2^{\omega}$ is a continuous map with a code in ${\mathfrak{M}}$, regular on $T_{0}$ inside ${\mathbf{P}}$, and $U,V\in\mathbf{Q}$, $U\subseteq T_{0}$, then $[V]\cap({f}\text{\rm\hskip 0.86108pt''}{[}U])=\varnothing$.∎

Speaking of (C), it claims essentially that if the regularity holds then $T_{0}$ forces, in the sense of $\mathbf{Q}\cup\mathbf{P}$, that $f(\mathbf{a}[G])$ does not belong to $\bigcup_{V\in\mathbf{Q}}[V]$.

If ${\mathfrak{M}}=\varnothing$ then we write $\mathbf{P}\mathrel{\sqsubset}\mathbf{Q}$ instead of $\mathbf{P}\mathrel{\sqsubset_{\varnothing}}\mathbf{Q}$; in this case (B) and (C) are trivial. Generally, we’ll consider, in the role of ${\mathfrak{M}}$, transitive models of the theory $\text{ZFC}^{\prime}$ which includes all ZFC axioms except for the Power Set axiom, but an axiom is adjoined, which claims the existence of ${\hskip 0.43057pt{\mathscr{P}}\hskip 0.43057pt}(\omega)$. (Then the existence of $\omega_{1}$ and sets like $2^{\omega}$ and $\mathbf{ST}$ easily follows.)

Lemma 8.2**.**

Let ${\mathfrak{M}}\models\text{ZFC}^{\prime}$ be a transitive model, $\mathbf{P}\in{\mathfrak{M}}$ and $\mathbf{Q}$ be ST-forcings satisfying $\mathbf{P}\mathrel{\sqsubset_{{\mathfrak{M}}}}\mathbf{Q}$, and $U\in\mathbf{Q}$. Then

1. (i)

if $T\in\mathbf{P}$ then $[U]\cap[T]$ is clopen in $[U]\,;$ 2. (ii)

if $D\in{\mathfrak{M}}$, $D\subseteq\mathbf{P}$ is pre-dense in $\mathbf{P}$ then $D$ remains pre-dense in $\mathbf{Q}\cup\mathbf{P}\,;$ 3. (iii)

if $T,T^{\prime}\in\mathbf{P}$ are incompatible in $\mathbf{P}$ then $T,T^{\prime}$ are incompatible in $\mathbf{Q}\cup\mathbf{P}$, too, and moreover if $U\in\mathbf{Q}$ then $[T]\cap[T^{\prime}]\cap[U]=\varnothing\;;$ 4. (iv)

if $R\subseteq U$ is a Silver tree then the set $\mathbf{R}=\mathbf{Q}\cup\{\hskip 0.43057pt{\sigma\boldsymbol{\cdot}(R{({\to\hskip 0.86108pt}t)})\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptt,\sigma\in 2^{<\omega}\hskip 0.43057pt\}$ is a ST-forcing, and still $\mathbf{P}\mathrel{\sqsubset_{{\mathfrak{M}}}}\mathbf{R}$.

Proof**.**

(i) The set $D(T)=\{\hskip 0.43057pt{S\in\mathbf{P}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptS\subseteq T\lor[S]\cap[T]=\varnothing\hskip 0.43057pt\}$ belongs to ${\mathfrak{M}}$ and is open dense in $\mathbf{P}$ by 6.1(A). Then by 8.1(B) there is a finite set $D^{\prime}\subseteq D(T)$ such that $U\subseteq\bigcup D^{\prime}$. But then we have $[U]\smallsetminus[T]=\bigcup_{S\in D^{\prime\prime}}([S]\cap[W])$, where $D^{\prime\prime}=\{\hskip 0.43057pt{S\in D^{\prime}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt[S]\cap[T]=\varnothing\hskip 0.43057pt\}$. Thus $[U]\smallsetminus[T]$ is a closed set.

(ii) Let $U\in\mathbf{Q}$. Then by 8.1(B) there is a finite set $D^{\prime}\subseteq D$ such that $U\subseteq\bigcup D^{\prime}$. There is a tree $T\in D^{\prime}$ such that $[T]\cap[U]$ has a non-empty open interior in $[U]$. By Lemma 4.3(i), there is $s\in U$ such that $[U^{\prime}]\subseteq[T]\cap[U]$, where $U^{\prime}={U}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507pts}$, thus $U^{\prime}\subseteq U\cap T$. Finally $U^{\prime}\in\mathbf{Q}$, as $\mathbf{Q}$ is a ST-forcing.

(iii) By the incompatibility, if $S\in\mathbf{P}$ then $S\not\subseteq T$ or $S\not\subseteq T^{\prime}$, and hence there is a tree $S^{\prime}\in\mathbf{P}$, $S^{\prime}\subseteq S$, satisfying $[S^{\prime}]\cap[T]\cap[T^{\prime}]=\varnothing$. Therefore the set $D=\{\hskip 0.43057pt{S\in\mathbf{P}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt[S]\cap[T]\cap[T^{\prime}]=\varnothing\hskip 0.43057pt\}$ is dense in $\mathbf{P}$ and belongs to ${\mathfrak{M}}$. Now if $U\in\mathbf{Q}$ then $U\subseteq^{\text{\tt fin}}\bigcup D$ by 8.1(B), thus immediately $[U]\cap[T]\cap[T^{\prime}]=\varnothing$.

(iv) $\mathbf{R}$ is a ST-forcing by Lemma 6.4, so we have to prove $\mathbf{P}\mathrel{\sqsubset_{{\mathfrak{M}}}}\mathbf{R}$. We skip a routine verification of 8.1(A),(B) and focus on (C). Let $T_{0}$, $f$ be as in (C).

Consider trees $R^{\prime}=\sigma\boldsymbol{\cdot}(R{({\to\hskip 0.86108pt}t)})\in{\hskip 0.43057pt{\mathbb{R}}\hskip 0.43057pt}$, $R^{\prime}\subseteq T_{0}$, and $V\in\mathbf{Q}$; we have to prove that $[V]\cap({f}\text{\rm\hskip 0.86108pt''}{[}R^{\prime}])=\varnothing$. The problem is that while $R^{\prime}\subseteq\sigma\boldsymbol{\cdot}R\subseteq U^{\prime}=\sigma\boldsymbol{\cdot}U$, it may not be true that $U^{\prime}\subseteq T_{0}$. But, now we claim that there is a finite set of trees $\{\hskip 0.43057ptU_{1},\dots,U_{n}\hskip 0.43057pt\}\subseteq\mathbf{Q}$ such that $R^{\prime}\subseteq U_{1}\cup\ldots\cup U_{n}\subseteq T_{0}$. If this is established then $[V]\cap({f}\text{\rm\hskip 0.86108pt''}{[}U_{i}])=\varnothing$ for all $i$ since $\mathbf{P}\mathrel{\sqsubset_{{\mathfrak{M}}}}\mathbf{Q}$, and we are done.

To prove the claim consider the dense set $D(T_{0})\in{\mathfrak{M}}$ as in the proof of (i) above. Then $U^{\prime}\subseteq^{\text{\tt fin}}\bigcup D$, hence there is a finite set $D^{\prime}\subseteq D(T_{0})$ satisfying $U^{\prime}\subseteq\bigcup D^{\prime}$. As by construction $R^{\prime}\subseteq U^{\prime}\cap T_{0}$, we conclude that $R^{\prime}\subseteq\bigcup D^{\prime\prime}$, where $D^{\prime\prime}=\{\hskip 0.43057pt{S\in D^{\prime}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptS\subseteq T_{0}\hskip 0.43057pt\}$. Now if $S\in D^{\prime\prime}$ then $[S]\cap[U^{\prime}]$ is clopen in $[U^{\prime}]$, therefore $[S]\cap[U^{\prime}]=\bigcup_{v\in W(S)}{U^{\prime}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptv}$, where $W(S)\subseteq U^{\prime}$ is a suitable finite set. We put $\{\hskip 0.43057ptU_{1},\dots,U_{n}\hskip 0.43057pt\}=\bigcup_{S\in D^{\prime\prime}}\{\hskip 0.43057pt{{U^{\prime}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptv}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptv\in W(S)\hskip 0.43057pt\}$. ∎

Theorem 8.3**.**

Let ${\mathfrak{M}}\models\text{ZFC}^{\prime}$ be a countable transitive model and $\mathbf{P}\in{\mathfrak{M}}$ be a ST-forcing. Then there exists a countable ST-forcing $\mathbf{Q}$ satisfying $\mathbf{P}\mathrel{\sqsubset_{{\mathfrak{M}}}}\mathbf{Q}$.

The proof of this theorem is presented below. Section 9 contains the definition of the ST-forcing $\mathbf{Q}$, while the proof of its properties follows in Section 10.

9 Construction of extending ST-forcing

The next definition formalizes a construction of countably many Silver trees by means of Lemma 5.3.

Definition 9.1**.**

A system is any indexed set $\varphi=\langle\langle\nu^{\varphi}_{m},\tau^{\varphi}_{m}\rangle\rangle_{m\in|\varphi|}$, where $|\varphi|\subseteq\omega$ is finite, and if $m\in|\varphi|$ then $\nu^{\varphi}_{m}\in\omega$, $\tau^{\varphi}_{m}=\langle T^{\varphi}_{m}(0),T^{\varphi}_{m}(1),\dots,T^{\varphi}_{m}(\nu^{\varphi}_{m})\rangle$, each $T^{\varphi}_{m}(n)$ is a Silver tree, and $T^{\varphi}_{m}(n+1)\mathrel{{\subseteq}_{n+1}}T^{\varphi}_{m}(n)$ whenever $n<\nu^{\varphi}_{m}$.

In this case, if $n\leq\nu^{\varphi}_{m}$ and $s\in 2^{n}$ then let $T^{\varphi}_{m}(s)=T^{\varphi}_{m}(n){({\to\hskip 0.86108pt}s)}$.

If $\mathbf{P}$ is a ST-forcing then $\text{Sys}(\mathbf{P})$ is the set of all system $\varphi$ such that $T^{\varphi}_{m}(n)\in\text{Colg}_{n}(\mathbf{P})$ for all $m\in|\varphi|$ and $n\leq\nu^{\varphi}_{m}$. Then every tree $T^{\varphi}_{m}(s)$ belongs to $\mathbf{P}$. ∎

A system $\varphi$ extends a system $\psi$, in symbol $\psi\preccurlyeq\varphi$, if $|\psi|\subseteq|\varphi|$, and for any index $m\in|\psi|$, first, $\nu^{\varphi}_{m}\geq\nu^{\psi}_{m}$, and second, $\tau^{\varphi}_{m}$ extends $\tau^{\psi}_{m}$ in the sense that simply $T^{\varphi}_{m}(n)=T^{\psi}_{m}(n)$ for all $n\leq\nu^{\psi}_{m}$.

Lemma 9.2**.**

Let $\mathbf{P}$ be a ST-forcing and $\varphi\in\text{Sys}(\mathbf{P})$.

If $m\in|\varphi|$ and $n=\nu^{\varphi}_{m}$ then the extension $\varphi^{\prime}$ of the system $\varphi$ by $\nu^{\varphi^{\prime}}_{m}=n+1$ and $T^{\varphi^{\prime}}_{m}(n+1)=T^{\varphi}_{m}(n)$ belongs to $\text{Sys}(\mathbf{P})$ and $\varphi\preccurlyeq\varphi^{\prime}$.

If $m\notin|\varphi|$ and $T\in\mathbf{P}$ then the extension $\varphi^{\prime}$ of the system $\varphi$ by $|\varphi^{\prime}|=|\varphi|\cup\{\hskip 0.43057ptm\hskip 0.43057pt\}$, $\nu^{\varphi^{\prime}}_{m}=0$ and $T^{\varphi^{\prime}}_{m}(0)=T$, belongs to $\text{Sys}(\mathbf{P})$ and $\varphi\preccurlyeq\varphi^{\prime}$.∎

Now, according to the formulation of Theorem 8.3, we assume that ${\mathfrak{M}}\models\text{ZFC}^{\prime}$ is a countable transitive model and $\mathbf{P}\in{\mathfrak{M}}$ is a (countable) ST-forcing.

Definition 9.3**.**

(i) We fix a $\hskip 0.86108pt{\preccurlyeq}$-increasing sequence $\mathbf{\Phi}=\langle\varphi(j)\rangle_{j<\omega}$ of systems $\varphi(j)\in\text{Sys}(\mathbf{P})$, generic over ${\mathfrak{M}}$ in the sense that it intersects every set $D\in{\mathfrak{M}}\hskip 0.43057pt,\linebreak[0]\>D\subseteq\text{Sys}(\mathbf{P})$ dense in $\text{Sys}(\mathbf{P})$. The density here means that for any system $\psi\in\text{Sys}(\mathbf{P})$ there is a system $\varphi\in D$ such that $\psi\subseteq\varphi$. The goal of this definition is to define another ST-forcing $\mathbf{Q}$ from $\mathbf{\Phi}$, see item (iv) below.

(ii) If $m,n<\omega$ then the set $D_{mn}=\{\hskip 0.43057pt{\varphi\in\text{Sys}(\mathbf{P})\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\nu^{\varphi}_{m}\geq n\hskip 0.43057pt\}$ is dense by Lemma 9.2 and belongs to ${\mathfrak{M}}$, hence it intersects $\mathbf{\Phi}$. Therefore if $m<\omega$ then there exists an infinite sequence

[TABLE]

of trees ${\mathbf{T}}^{\mathbf{\Phi}}_{m}(n)\in\text{Colg}_{n}(\mathbf{P})$ such that for each $j$, if $m\in|\varphi(j)|$ and $n\leq\nu^{\varphi(j)}_{m}$, then $T^{\varphi(j)}_{m}(n)={\mathbf{T}}^{\mathbf{\Phi}}_{m}(n)$. If $n<\omega$ and $s\in 2^{n}$ then let ${\mathbf{T}}^{\mathbf{\Phi}}_{m}(s)={\mathbf{T}}^{\mathbf{\Phi}}_{m}(n){({\to\hskip 0.86108pt}s)}$; then ${\mathbf{T}}^{\mathbf{\Phi}}_{m}(s)\in\mathbf{P}$ because ${\mathbf{T}}^{\mathbf{\Phi}}_{m}(n)\in\text{Colg}_{n}(\mathbf{P})$.

(iii) In this case, by Lemma 5.3, every set

[TABLE]

belongs to $\mathbf{ST}$, the same is true for all subtrees ${\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}s)}$, and we have

[TABLE]

by Lemma 5.3, and it is clear that ${\mathbf{U}}^{\mathbf{\Phi}}_{m}={\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}\Lambda)}$. In addition if strings $s,t\in 2^{<\omega}$ satisfy $s\subseteq t$ then ${\mathbf{T}}^{\mathbf{\Phi}}_{m}(s)\subseteq{\mathbf{T}}^{\mathbf{\Phi}}_{m}(t)$ and ${\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}s)}\subseteq{\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}t)}$, but if $s,t$ are $\hskip 0.86108pt{\subseteq}$-incomparable then $[{\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}s)}]\cap[{\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}t)}]=[{\mathbf{T}}^{\mathbf{\Phi}}_{m}(s)]\cap[{\mathbf{T}}^{\mathbf{\Phi}}_{m}(t)]=\varnothing$.

(iv) Finally the set $\mathbf{Q}=\{\hskip 0.43057pt{\sigma\boldsymbol{\cdot}{\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}s)}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptm<\omega\land\sigma,s\in 2^{<\omega}\hskip 0.43057pt\}$ is a countable ST-forcing by Lemma 6.4. ∎

10 Validation of the extension property

Here we prove that $\mathbf{P}\mathrel{\sqsubset_{{\mathfrak{M}}}}\mathbf{Q}$ in the context of Definition 9.3. We check all requirements of Definition 8.1.

8.1**(A)****.** If $T\in\mathbf{P}$ then the set $\Delta(T)$ of all systems $\varphi\in\text{Sys}(\mathbf{P})\cap{\mathfrak{M}}$ such that $T^{\varphi}_{m}(0)=T$ for some $m$, belongs to ${\mathfrak{M}}$ and is dense in $\text{Sys}(\mathbf{P})$ by Lemma 9.2. Therefore $\varphi(J)\in\Delta(T)$ for some $J$, by the choice of $\mathbf{\Phi}$. Then ${\mathbf{T}}^{\mathbf{\Phi}}_{m}(0)=T$ for some $m<\omega$. But ${\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}\Lambda)}={\mathbf{U}}^{\mathbf{\Phi}}_{m}\subseteq{\mathbf{T}}^{\mathbf{\Phi}}_{m}(0)$ and ${\mathbf{U}}^{\mathbf{\Phi}}_{m}\in\mathbf{Q}{}$.

8.1**(B)****.** Let $U=\sigma\boldsymbol{\cdot}{\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}s)}\in\mathbf{Q}$, where $m<\omega$ and $s,\sigma\in 2^{<\omega}.$ We have to check $U\subseteq^{\text{\tt fin}}\bigcup D$. We can assume, as above, that $\sigma=\Lambda$, that is, $U={\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}s)}$. (Otherwise substitute $\sigma\boldsymbol{\cdot}D$ for $D$.) We can also assume that $s=\Lambda$, that is, $U={\mathbf{U}}^{\mathbf{\Phi}}_{m}$, as ${\mathbf{U}}^{\mathbf{\Phi}}_{m}{({\to\hskip 0.86108pt}s)}\subseteq{\mathbf{U}}^{\mathbf{\Phi}}_{m}$. Thus let $U={\mathbf{U}}^{\mathbf{\Phi}}_{m}$.

The set $\Delta\in{\mathfrak{M}}$ of all systems $\varphi\in\text{Sys}(\mathbf{P})$ such that $m\in|\varphi|$ and for every string $t\in 2^{n}$, where $n=\nu^{\varphi}_{m}$, there is a tree $S_{t}\in D$ with $T^{\varphi}_{m}(t)\subseteq S_{t}$, is dense in $\text{Sys}(\mathbf{P})$ by lemma 6.6(iv) due to the pre-density of $D$ itself. Therefore there is an index $j$ such that $\varphi(j)\in\Delta$. Let this be witnessed by trees $S_{t}\in D\hskip 0.43057pt\text{,}\linebreak[0]\;t\in 2^{n},$ where $n=\nu^{\varphi(j)}_{m}$, so that $T^{\varphi(j)}_{m}(t)\subseteq S_{t}$, ${\forall\,}t$, and hence $T^{\varphi(j)}_{m}(n)\subseteq^{\text{\tt fin}}D$. Then $\textstyle U\,=\,{\mathbf{U}}^{\mathbf{\Phi}}_{m}\,\subseteq\,{T}^{\mathbf{\Phi}}_{m}(n)\,=\,T^{\varphi(j)}_{m}(n)\,\subseteq^{\text{\tt fin}}\,\bigcup D\,.$

8.1**(C)****.** Let $T_{0}\in\mathbf{P}$, $f$, and $U,V\in\mathbf{Q}$ be as in 8.1(C). Then $U=\tau\boldsymbol{\cdot}{\mathbf{U}}^{\mathbf{\Phi}}_{K}{({\to\hskip 0.86108pt}s_{0})}$, where $s_{0},\tau\in 2^{<\omega}$ and $K<\omega$. We can wlog assume that $\tau=\Lambda$, that is, $U={\mathbf{U}}^{\mathbf{\Phi}}_{K}{({\to\hskip 0.86108pt}s_{0})}$, since the general case is reducible to this case by the substitution of $\tau\boldsymbol{\cdot}T_{0}$ for $T_{0}$ and the function $f^{\prime}(x)=f(\tau\boldsymbol{\cdot}x)$ for $f$. Thus let $U={\mathbf{U}}^{\mathbf{\Phi}}_{K}{({\to\hskip 0.86108pt}s_{0})}$.

Similarly, generally speaking $V=\rho\boldsymbol{\cdot}{\mathbf{U}}^{\mathbf{\Phi}}_{L}{({\to\hskip 0.86108pt}t_{0})}$, where $t_{0},\rho\in 2^{<\omega}$ and $L<\omega$. But this is reducible to the case $\rho=\Lambda$ by the substitution of $f^{\prime}(x)=\rho\boldsymbol{\cdot}f(x)$ for $f$. Thus let $V={\mathbf{U}}^{\mathbf{\Phi}}_{L}{({\to\hskip 0.86108pt}t_{0})}$. Finally, since ${\mathbf{U}}^{\mathbf{\Phi}}_{L}{({\to\hskip 0.86108pt}t_{0})}\subseteq{\mathbf{U}}^{\mathbf{\Phi}}_{L}$, we can assume that even $V={\mathbf{U}}^{\mathbf{\Phi}}_{L}$. Now consider the set $\Delta\in{\mathfrak{M}}$ of all systems $\varphi\in\text{Sys}(\mathbf{P})$ such that there is a number $m<\omega$ satisfying the following:

1. (I)

$K,L\in|\varphi|$, $\nu^{\varphi}_{K}=\nu^{\varphi}_{L}=m$, and $\mathop{\tt lh}(s_{0})\leq m$; 2. (II)

if $s\in 2^{m}$ then $T^{\varphi}_{K}(s)\subseteq T_{0}$ or $[T^{\varphi}_{K}(s)]\cap[T_{0}]=\varnothing$; 3. (III)

if $s\in 2^{m}$ and $T^{\varphi}_{K}(s)\subseteq T_{0}$ then $T^{\varphi}_{L}(m)\cap({f}\text{\rm\hskip 0.86108pt''}{[}T^{\varphi}_{K}(s)])=\varnothing$.

Lemma 10.1**.**

The set $\Delta$ is dense in $\text{Sys}(\mathbf{P})$.

Proof**.**

Let $\psi\in\text{Sys}(\mathbf{P})$; we have to define a system $\varphi\in\Delta$ satisfying $\psi\preccurlyeq\varphi$. By lemma 9.2, we can assume that $K,L\in|\psi|$ and $\nu^{\psi}_{K}=\nu^{\psi}_{L}=m-1$ for some $m\geq\mathop{\tt lh}(s_{0})$. Now we define a first preliminary version of $\varphi$, by $\nu^{\varphi}_{K}=\nu^{\psi}_{L}=m$ and $T^{\varphi}_{K}(m)=T^{\psi}_{K}(m-1)$, $T^{\varphi}_{L}(m)=T^{\psi}_{L}(m-1)$, and keeping the other elements of $\varphi$ equal to those of the system $\psi$, so that $\psi\preccurlyeq\varphi$. The trees $S=T^{\varphi}_{K}(m)$ and $T=T^{\varphi}_{L}(m)$ belong to $\text{Colg}_{m}(\mathbf{P})$.

The set $D(T_{0})$ of all trees $W\in\mathbf{P}$, such that $W\subseteq T_{0}$ or $[W]\cap[T_{0}]=\varnothing$, is open dense in $\mathbf{P}$ by 6.1(A). Therefore by Lemma 6.6(iv) there is a tree $S^{\prime}\in\text{Colg}_{m}(\mathbf{P})$ satisfying $S^{\prime}\mathrel{{\subseteq}_{m}}S$ and if $s\in 2^{m}$ then $S^{\prime}{({\to\hskip 0.86108pt}s)}\in D(T_{0})$. We modify $\varphi$ by putting $T^{\varphi}_{K}(m)=S^{\prime}$ instead of $S$. It is clear that the new system still satisfies $\psi\preccurlyeq\varphi$, and in addition (II) holds by construction.

Further modification of $\varphi$ towards (III) depends on whether $K=L$.

Case 1: $K\neq L$. If $s,t\in 2^{m}$ then by Lemma 7.2(i) there exist trees $C,D\in\mathbf{P}$ such that $C\subseteq S{({\to\hskip 0.86108pt}s)}$, $D\subseteq T{({\to\hskip 0.86108pt}t)}$, and $[D]\cap({f}\text{\rm\hskip 0.86108pt''}{[C]})=\varnothing$. Applying Lemma 6.6(iii), we get trees $S^{\prime},T^{\prime}\in\text{Colg}_{m}(\mathbf{P})$ satisfying $S^{\prime}\mathrel{{\subseteq}_{m}}S$, $T^{\prime}\mathrel{{\subseteq}_{m}}T$, $C=S^{\prime}{({\to\hskip 0.86108pt}s)}$, $D=T^{\prime}{({\to\hskip 0.86108pt}t)}$, so that $[T^{\prime}{({\to\hskip 0.86108pt}t)}]\cap({f}\text{\rm\hskip 0.86108pt''}{[S^{\prime}{({\to\hskip 0.86108pt}s)}]})=\varnothing$. Iterate this construction by exhaustion of all pairs $s,t\in 2^{m}$. We get trees $S^{\ast},T^{\ast}\in\text{Colg}_{m}(\mathbf{P})$ such that $S^{\ast}\mathrel{{\subseteq}_{m}}S$, $T^{\ast}\mathrel{{\subseteq}_{m}}T$, and $[T^{\ast}{({\to\hskip 0.86108pt}t)}]\cap({f}\text{\rm\hskip 0.86108pt''}{[S^{\ast}{({\to\hskip 0.86108pt}s)}]})=\varnothing$ for all $s,t\in 2^{m}$, that is $[{T^{\ast}}]\cap({f}\text{\rm\hskip 0.86108pt''}{[S^{\ast}]})=\varnothing$. Nodify $\varphi$ by $T^{\varphi}_{K}(m)=S^{\ast}$ instead of $S$ and $T^{\varphi}_{L}(m)=T^{\ast}$ instead of $T$. Now (III) holds for the modified $\varphi$.

Case 2: $K=L$, and then $S=T=T^{\varphi}_{K}(m)=T^{\varphi}_{L}(m)$. It suffices to show that if $s,t\in 2^{n}$ (strings of length $n$) and $S{({\to\hskip 0.86108pt}s)}\subseteq T_{0}$, then there is a tree $S^{\prime}\in\text{Colg}_{n}(P)$ such that $S^{\prime}\mathrel{{\subseteq}_{m}}S$ and $[S^{\prime}{({\to\hskip 0.86108pt}t)}]\cap({f}\text{\rm\hskip 0.86108pt''}{[}S^{\prime}{({\to\hskip 0.86108pt}s)}])=\varnothing$. Then the iteration by exhaustion of all those pairs of strings $s,t$ yields a tree $S^{\ast}\in\text{Colg}_{m}(\mathbf{P})$ such that $S^{\ast}\mathrel{{\subseteq}_{m}}S$ and $[T^{\ast}{({\to\hskip 0.86108pt}t)}]\cap({f}\text{\rm\hskip 0.86108pt''}{[S^{\ast}{({\to\hskip 0.86108pt}s)}]})=\varnothing$ for all $s,t\in 2^{m}$ with $S{({\to\hskip 0.86108pt}s)}\subseteq T_{0}$, that is, $[{S^{\ast}}]\cap({f}\text{\rm\hskip 0.86108pt''}{[S^{\ast}{({\to\hskip 0.86108pt}s)}]})=\varnothing$ whenevew $s\in 2^{m}$ satisfies $S{({\to\hskip 0.86108pt}s)}\subseteq T_{0}$. To achieve (III), it remains to modify the system $\varphi$ by $T^{\varphi}_{K}(m)=S^{\ast}$ instead of $S$.

Thus let us carry out the construction of $S^{\prime}$ for a pair of strings $s,t\in 2^{n}$ with $S{({\to\hskip 0.86108pt}s)}\subseteq T_{0}$. It follows that $f$ is regular on $S{({\to\hskip 0.86108pt}s)}$ inside $\mathbf{P}$. By Lemma 5.1(i), we have $S{({\to\hskip 0.86108pt}s)}={S}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}$ and $S{({\to\hskip 0.86108pt}t)}={S}{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptv}$, where $u,v\in T$ are strings of length $h=\mathop{\tt lh}(u)=\mathop{\tt lh}(v)$, and $S{({\to\hskip 0.86108pt}s)}=\tau\boldsymbol{\cdot}(S{({\to\hskip 0.86108pt}t)})$, where $\tau=u\boldsymbol{\cdot}v$, by Lemma 4.3(ii). Lemma 7.2(ii) yields a pair of trees $U,V\in\mathbf{P}$ satisfying $U\subseteq S{({\to\hskip 0.86108pt}s)}$, $V\subseteq S{({\to\hskip 0.86108pt}t)}$, $V=\tau\boldsymbol{\cdot}U$, and $[V]\cap({f}\text{\rm\hskip 0.86108pt''}{[}U])=\varnothing$. Now, twice reducing the tree $S$ by means of Lemma 6.6(iii), we get a tree $S^{\prime}\in\text{Colg}_{n}(P)$ such that $S^{\prime}\mathrel{{\subseteq}_{m}}S$ and $S^{\prime}{({\to\hskip 0.86108pt}s)}=U$, $S^{\prime}{({\to\hskip 0.86108pt}t)}=V$, so that $[S^{\prime}{({\to\hskip 0.86108pt}t)}]\cap({f}\text{\rm\hskip 0.86108pt''}{[}S^{\prime}{({\to\hskip 0.86108pt}s)}])=\varnothing$, as required. $\square$ (Lemma)

Now return to the verification of 8.1(C). By the lemma, at least one system $\varphi(j)$ belongs to $\Delta$, that is, conditions (I), (II), (III) are satisfied for $\varphi=\varphi(j)$. The tree $V={\mathbf{U}}^{\mathbf{\Phi}}_{L}$ satisfies $V\subseteq T=T^{\varphi(j)}_{L}(m)$. Moreover, the tree $U={\mathbf{U}}^{\mathbf{\Phi}}_{K}{({\to\hskip 0.86108pt}s_{0})}$ satisfies $U\subseteq S=\bigcup_{s\in\Sigma}T^{\varphi(j)}_{K}(s)$, where $\Sigma=\{\hskip 0.43057pt{s\in 2^{m}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptS{({\to\hskip 0.86108pt}s)}\subseteq T_{0}\hskip 0.43057pt\}$, by (I), (II). However $[T]\cap({f}\text{\rm\hskip 0.86108pt''}{[}S])=\varnothing$ by (III), thus $[V]\cap({f}\text{\rm\hskip 0.86108pt''}{[}U])=\varnothing$, as required.

$\square$ (Theorem 8.3)

11 The blocking sequence of ST-forcings

We argue in the constructible universe ${\mathbf{L}}$ in this section.

The forcing to prove Theorem 1.1 will be defined as the union of a $\hskip 0.86108pt{\omega_{1}}$-sequence of countable ST-forcings, increasing in the sense of a relation $\mathrel{\sqsubset}$ (Definition 8.1). We here introduce the notational system to be used in this construction.

Let $\mathbf{STF}$ be the set of all countable ST-forcings.

If $\overline{\mathbf{P}}=\langle\mathbf{P}_{\xi}\rangle_{\xi<\omega_{1}}$ is a transfinite sequence of countable ST-forcings, of length $\mathop{\tt dom}\overline{\mathbf{P}}=\lambda<\omega_{1}$, then let $\bigcup\overline{\mathbf{P}}=\bigcup_{\xi<\lambda}\mathbf{P}_{\xi}$, and let ${\mathfrak{M}}(\overline{\mathbf{P}})$ be the least transitive model of $\text{ZFC}^{-}$ of the form ${\mathbf{L}}_{\vartheta}$, containing $\overline{\mathbf{P}}$, in which $\lambda$ and $\bigcup\overline{\mathbf{P}}$ are countable.

Definition 11.1**.**

If $\alpha\in\text{Ord}$ then let $\overline{\mathbf{STF}}_{\alpha}$ be the set of all $\hskip 0.86108pt{\lambda}$-sequences $\overline{\mathbf{P}}=\langle\mathbf{P}_{\xi}\rangle_{\xi<\alpha}$ of forcings $\mathbf{P}_{\xi}\in\mathbf{STF}$, satisfying the following:

1. $(*)$

if  $\gamma<\mathop{\tt dom}\overline{\mathbf{P}}$  then  ${\bigcup{(\overline{\mathbf{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\gamma)}}\mathrel{\sqsubset_{{\mathfrak{M}}(\overline{\mathbf{P}}{\hskip 0.3014pt\restriction\hskip 0.90417pt}\gamma)}}\mathbf{P}_{\gamma}$.

Let $\overline{\mathbf{STF}}=\bigcup_{\alpha<\omega_{1}}\overline{\mathbf{STF}}_{\alpha}$. ∎

The set $\overline{\mathbf{STF}}\cup\overline{\mathbf{STF}}_{\omega_{1}}$ is ordered by the extension relations $\subset$ and $\subseteq$.

Lemma 11.2**.**

Assume that $\kappa<\lambda<\omega_{1}$, and $\overline{\mathbf{P}}=\langle\mathbf{P}_{\xi}\rangle_{\xi<\kappa}\in\overline{\mathbf{STF}}$. Then$:$

1. (i)

the union $\mathbf{P}=\bigcup\overline{\mathbf{P}}$ is a countable ST-forcing*;*** 2. (ii)

there is a sequence $\overline{\mathbf{Q}}\in\overline{\mathbf{STF}}$ such that $\mathop{\tt dom}(\overline{\mathbf{Q}})=\lambda$ and $\overline{\mathbf{P}}\subset\overline{\mathbf{Q}}\;.$

Proof**.**

To prove (ii) apply Theorem 8.3 by induction on $\lambda$. ∎

Definition 11.3** (key definition).**

A sequence $\overline{\mathbf{P}}\in\overline{\mathbf{STF}}$ blocks a set $W\subseteq\overline{\mathbf{STF}}$ if either $\overline{\mathbf{P}}\in W$ (the positive block case) or there is no sequence $\overline{\mathbf{Q}}\in W$ satisfying $\overline{\mathbf{P}}\subseteq\overline{\mathbf{Q}}$ (the negative block case). ∎

Approaching the next blocking sequence theorem, we recall that $\mathrm{HC}$ is the set of all hereditarily countable sets, so that $\mathrm{HC}={\mathbf{L}}_{\omega_{1}}$ in ${\mathbf{L}}$. See [1, Part B, Chap. 5, Section 4] on definability classes $\varSigma^{X}_{n},\,\varPi^{X}_{n},\,\varDelta^{X}_{n}$ for any set $X$, and especially on $\varSigma^{\mathrm{HC}}_{n},\,\varPi^{\mathrm{HC}}_{n},\,\varDelta^{\mathrm{HC}}_{n}$ for $X=\mathrm{HC}$ in [12, Sections 8, 9] or elsewhere.

Theorem 11.4** (blocking sequence theorem, in ${\mathbf{L}}$).**

If $\mathbb{n}\geq 3$ then there is a sequence $\overline{\mathbb{P}}=\langle{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}\rangle_{\xi<\omega_{1}}\in\overline{\mathbf{STF}}_{\omega_{1}}$ satisfying the following two conditions$:$

1. (i)

$\overline{\mathbb{P}}$, as the set of pairs $\langle\xi,{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}\rangle$, belongs to the definability class $\varDelta^{\mathrm{HC}}_{\mathbb{n}-1}\;;$ 2. (ii)

(genericity of $\overline{\mathbf{P}}$ w. r. t. $\varSigma^{\mathrm{HC}}_{\mathbb{n}-2}(\mathrm{HC})$ sets)*  if $W\subseteq\overline{\mathbf{STF}}$ is a $\varSigma^{\mathrm{HC}}_{\mathbb{n}-2}(\mathrm{HC})$ set (that is parameters from $\mathrm{HC}$ are admitted), *then there is an ordinal $\gamma<\omega_{1}$ such that the restricted sequence $\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\gamma=\langle{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}\rangle_{\xi<\gamma}\in\overline{\mathbf{STF}}$ blocks $W$.

Proof**.**

Let $\leqslant_{{\mathbf{L}}}$ be the canonical $\varDelta_{1}$ wellordering of ${\mathbf{L}}$; thus its restriction to $\mathrm{HC}={\mathbf{L}}_{\omega_{1}}$ is $\varDelta^{\mathrm{HC}}_{1}$. As $\mathbb{n}\geq 3$, there exists a universal $\varSigma^{\mathrm{HC}}_{\mathbb{n}-2}$ set ${\hskip 0.43057pt{\mathfrak{U}}\hskip 0.43057pt}\subseteq\omega_{1}\times\mathrm{HC}$. That is, ${\hskip 0.43057pt{\mathfrak{U}}\hskip 0.43057pt}$ is $\varSigma^{\mathrm{HC}}_{\mathbb{n}-2}$ (parameter-free $\varSigma_{\mathbb{n}-2}$ definable in $\mathrm{HC}$), and for every set $X\subseteq\mathrm{HC}$ of class $\varSigma^{\mathrm{HC}}_{\mathbb{n}-2}(\mathrm{HC})$ (definable in $\mathrm{HC}$ by a $\varSigma_{\mathbb{n}-2}$ formula with arbitrary parameters in $\mathrm{HC}$) there is an ordinal $\alpha<\omega_{1}$ such that $X={\hskip 0.43057pt{\mathfrak{U}}\hskip 0.43057pt}_{\alpha}$, where ${\hskip 0.43057pt{\mathfrak{U}}\hskip 0.43057pt}_{\alpha}=\{\hskip 0.43057pt{x\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\langle\alpha,x\rangle\in{\hskip 0.43057pt{\mathfrak{U}}\hskip 0.43057pt}\hskip 0.43057pt\}$). The choice of $\omega_{1}$ as the domain of parameters is validated by the assumption ${\mathbf{V}}={\mathbf{L}}$, which implies the existence of a $\varDelta^{\mathrm{HC}}_{1}$ surjection $\omega_{1}\overset{{\text{onto}}}{\longrightarrow}\mathrm{HC}$.

Coming back to Definition 11.3, note that for any sequence $\overline{\mathbf{P}}\in\overline{\mathbf{STF}}$ and any set $W\subseteq\overline{\mathbf{STF}}$ there is a sequence $\overline{\mathbf{Q}}\in\overline{\mathbf{STF}}$ which satisfies $\overline{\mathbf{P}}\subset\overline{\mathbf{Q}}$ and blocks $W$. This allows us to define $\overline{\mathbf{Q}}_{\alpha}\in\overline{\mathbf{STF}}$ by induction on $\alpha<\omega_{1}$ so that $\overline{\mathbf{Q}}_{0}=\varnothing$, $\overline{\mathbf{Q}}_{\lambda}=\bigcup_{\alpha<\lambda}\overline{\mathbf{Q}}_{\alpha}$, and each $\overline{\mathbf{Q}}_{\alpha+1}$ is equal to the $\hskip 0.86108pt{\leqslant_{{\mathbf{L}}}}$-least sequence $\overline{\mathbf{Q}}\in\overline{\mathbf{STF}}$ which satisfies $\overline{\mathbf{Q}}_{\alpha}\subset\overline{\mathbf{Q}}$ and blocks ${\hskip 0.43057pt{\mathfrak{U}}\hskip 0.43057pt}_{\alpha}$. Then $\overline{\mathbb{P}}=\bigcup_{\alpha<\omega_{1}}\overline{\mathbf{Q}}_{\alpha}\in\overline{\mathbf{STF}}_{\omega_{1}}$.

Condition (ii) holds by construction, while (i) follows by a routine verification, based on the fact that $\overline{\mathbf{STF}}\in\varDelta^{\mathrm{HC}}_{1}$. ∎

Definition 11.5** (in ${\mathbf{L}}$).**

We fix a natural number $\mathbb{n}\geq 3$, for which Theorem 1.1 is to be established. We also fix a sequence $\overline{\mathbb{P}}=\langle{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}\rangle_{\xi<\omega_{1}}\in\overline{\mathbf{STF}}_{\omega_{1}}$, given by Theorem 11.4 for this $\mathbb{n}$.

If $\alpha<\omega_{1}$ then let ${\mathfrak{M}}_{\alpha}={\mathfrak{M}}(\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\alpha)$

and ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}=\bigcup_{\xi<\alpha}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}$.

Let ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}=\bigcup_{\xi<\omega_{1}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}$. ∎

12 CCC and some other forcing properties

The ST-forcing ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ defined by 11.5 will be the forcing notion for the proof of Theorem 1.1. Here we establish some forcing properties of ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$, including CCC.

We continue to argue in the conditions and notation of Definition 11.5.

Lemma 12.1**.**

${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$* and all sets ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\xi}$, ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ are ST-forcings. In addition$:$*

1. (i)

if $\alpha<\omega_{1}$ then ${{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\gamma}}\mathrel{\sqsubset_{{\mathfrak{M}}_{\gamma}}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\gamma}\,;$ 2. (ii)

*if $\alpha<\omega_{1}$ and the set

$D\in{\mathfrak{M}}_{\alpha}\hskip 0.43057pt\text{,}\linebreak[0]\;D\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ is pre-dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ then it is pre-dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$, too**;*** 3. (iii)

every set ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ is pre-dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}\,;$ 4. (iv)

if $Q\subseteq\mathbf{ST}$ belongs to $\varSigma^{\mathrm{HC}}_{\mathbb{n}-2}(\mathrm{HC})$ and $Q^{-}=\{\hskip 0.43057pt{T\in\mathbf{ST}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\neg\>{\exists\,}S\in Q\,(S\subseteq T)\hskip 0.43057pt\}$ then ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}\cap{(Q\cup Q^{-})}$ is dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$;** 5. (v)

if $\mathbf{c}\in\text{CCF}$ is a code of continuous function and

[TABLE]

then the set ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}\cap{\text{CB}_{\mathbf{c}}}$ is dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$;** 6. (vi)

let $D=\{\hskip 0.43057pt{T\in\mathbf{ST}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\text{\tt spl}(T)\text{ is co-infinite}\hskip 0.43057pt\}$ (see Definition 4.1 on $\text{\tt spl}(T)$), then the set ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}\cap D$ is dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$.

Proof**.**

(i) holds by $(*)$ ‣ 11.1 of Definition 11.1.

(ii) We use induction on $\gamma\hskip 0.43057pt,\linebreak[0]\>\alpha\leq\gamma<\omega_{1}$, to check that if $D$ is pre-dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\gamma}$ then it remains pre-dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\gamma}\cup{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\gamma}={\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\gamma+1}$ by (i) and 8.1(B). Limit steps, including the final step to ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ ($\gamma=\omega_{1}$) are routine.

(iii) ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ is dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha+1}={\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}\cup{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ by 8.1(A). It remains to refer to (ii).

(iv) Let $T_{0}\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$, that is, $T_{0}\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha_{0}}$, $\alpha_{0}<\omega_{1}$. The set $W$ of all sequences $\overline{\mathbf{P}}\in\overline{\mathbf{STF}}$, such that $\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\alpha_{0}\subseteq\overline{\mathbf{P}}$ and ${\exists\,}T\in Q\cap(\bigcup\overline{\mathbf{P}})\,(T\subseteq T_{0})$, belongs to $\varSigma^{\mathrm{HC}}_{\mathbb{n}-2}(\mathrm{HC})$ along with $Q$. Therefore there is an ordinal $\alpha<\omega_{1}$ such that $\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\alpha$ blocks $W$. We have two cases.

Case 1: $\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\alpha\in W$. Then the related tree $T\subseteq T_{0}$ belongs to $Q\cap{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$.

Case 2: there is no sequence in $W$ which extends $\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\alpha$. Let $\gamma=\mathop{\tt max}\{\hskip 0.43057pt\alpha,\alpha_{0}\hskip 0.43057pt\}$. Then ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\gamma}\mathrel{\sqsubset_{{\mathfrak{M}}_{\gamma}}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\gamma}$ by (i). As $\alpha_{0}<\gamma$, there is a tree $T\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\gamma}$, $T\subseteq T_{0}$. We claim that $T\in Q^{-}$, which completes the proof in Case 2.

Suppose to the contrary that $T\notin Q^{-}$, thus there is a tree $S\in Q$, $S\subseteq T$. The set $\mathbf{R}={\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\gamma}\cup\{\hskip 0.43057pt{\sigma\boldsymbol{\cdot}(S{({\to\hskip 0.86108pt}t)})\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptt,\sigma\in 2^{<\omega}\hskip 0.43057pt\}$ is a countable ST-forcing and ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\gamma}\mathrel{\sqsubset_{{\mathfrak{M}}_{\gamma}}}\mathbf{R}$ by Lemma 8.2(iv). It follows that the sequence $\overline{\mathbf{R}}$ defined by $\mathop{\tt dom}\overline{\mathbf{R}}=\gamma+1$, $\overline{\mathbf{R}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\gamma=\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\gamma$, and $\overline{\mathbf{R}}(\gamma)=\mathbf{R}$, belongs to $\overline{\mathbf{STF}}$, and even $\overline{\mathbf{R}}\in W$ since $S\in Q\cap\mathbf{R}$. Yet $\overline{\mathbb{P}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}\alpha\mathrel{\sqsubset_{{\mathfrak{M}}_{\gamma}}}\overline{\mathbf{R}}$ by construction, which contradicts to the Case 2 hypothesis.

(v) A routine estimation gives $\text{CB}_{\mathbf{c}}\in\varSigma^{\mathrm{HC}}_{1}(\{\hskip 0.43057pt\mathbf{c}\hskip 0.43057pt\})$. The set $\text{CB}_{\mathbf{c}}$ is dense in $\mathbf{ST}$ by Corollary 5.4, thus $(\text{CB}_{\mathbf{c}})^{-}$ is empty. Now the result follows from (iv).

(vi) Similarly to (v), $D\in\varSigma^{\mathrm{HC}}_{1}$ and $D$ is dense in $\mathbf{ST}$. ∎

Corollary 12.2**.**

If $\alpha<\omega_{1}$ and trees $T,T^{\prime}\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ are incompatible in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ then $T,T^{\prime}$ are incompatible in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$, too.

Proof**.**

Prove by induction on $\gamma$ that if $\alpha<\gamma\leq\omega_{1}$ then $T,T^{\prime}$ are incompatible in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\gamma}$, using Lemma 12.1(i), and Lemma 8.2(iii) on limit steps. ∎

To prove CCC we’ll need the following lemma.

Lemma 12.3** (in ${\mathbf{L}}$).**

If $X\subseteq\mathrm{HC}={\mathbf{L}}_{\omega_{1}}$ then the set $\mathscr{O}_{X}$ of all ordinals $\alpha<\omega_{1}$ such that the model $\langle{\mathbf{L}}_{\alpha}\hskip 1.29167pt;\hskip 0.43057ptX\cap{\mathbf{L}}_{\alpha}\rangle$ is an elementary submodel of $\langle{\mathbf{L}}_{\omega_{1}}\hskip 1.29167pt;\hskip 0.43057ptX\rangle$ and $X\cap{\mathbf{L}}_{\alpha}\in{\mathfrak{M}}_{\alpha}$, is unbounded in $\omega_{1}$. Generally if $X_{n}\subseteq\mathrm{HC}$ for all $n$ then the set $\mathscr{O}$ of all ordinals $\alpha<\omega_{1}$ such that $\langle{\mathbf{L}}_{\alpha}\hskip 1.29167pt;\hskip 0.43057pt\langle X_{n}\cap{\mathbf{L}}_{\alpha}\rangle_{n<\omega}\rangle$ is an elementary submodel of $\langle{\mathbf{L}}_{\omega_{1}}\hskip 1.29167pt;\hskip 0.43057pt\langle X_{n}\rangle_{n<\omega}\rangle$ and $\langle X_{n}\cap{\mathbf{L}}_{\alpha}\rangle_{n<\omega}\in{\mathfrak{M}}_{\alpha}$, is unbounded in $\omega_{1}$.

Proof**.**

Let $\alpha_{0}<\omega_{1}$. There is a countable elementary submodel $M$ of $\langle{\mathbf{L}}_{\omega_{2}}\hskip 1.29167pt;\hskip 0.43057pt\in\rangle$ which contains $\alpha_{0}\hskip 0.43057pt,\linebreak[0]\hskip 0.43057pt\omega_{1}\hskip 0.43057pt,\linebreak[0]\hskip 0.43057ptX$ and is such that the set $M\cap{\mathbf{L}}_{\omega_{1}}$ is transitive. Consider the Mostowski collapse $\phi:M\overset{{\text{onto}}}{\longrightarrow}{\mathbf{L}}_{\lambda}$. Let $\alpha=\phi(\omega_{1})$. Then $\alpha_{0}<\alpha<\lambda<\omega_{1}$ and $\phi(X)=X\cap{\mathbf{L}}_{\alpha}$ by the choice of $M$. We conclude that $\langle{\mathbf{L}}_{\alpha}\hskip 1.29167pt;\hskip 0.43057ptX\cap{\mathbf{L}}_{\alpha}\rangle$ is an elementary submodel of $\langle{\mathbf{L}}_{\omega_{1}}\hskip 1.29167pt;\hskip 0.43057ptX\rangle$. And $\alpha$ is uncountable in ${\mathbf{L}}_{\lambda}$, hence ${\mathbf{L}}_{\lambda}\subseteq{\mathfrak{M}}_{\alpha}$. It follows that $X\cap{\mathbf{L}}_{\alpha}\in{\mathfrak{M}}_{\alpha}$, as $X\cap{\mathbf{L}}_{\alpha}\in{\mathbf{L}}_{\lambda}$ by construction.

The more general claim is proved similarly. ∎

Corollary 12.4** (in ${\mathbf{L}}$).**

${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$* is a CCC forcing, therefore $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic extensions preserve cardinals.*

Proof**.**

Consider any maximal antichain $A\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$. By Lemma 12.3 there is an orrdinal $\alpha$ such that $\langle{\mathbf{L}}_{\alpha}\hskip 1.29167pt;\hskip 0.43057pt{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}^{\prime},A^{\prime}\rangle$ is an elementary submodel of $\langle{\mathbf{L}}_{\omega_{1}}\hskip 1.29167pt;\hskip 0.43057pt{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt},A\rangle$, where ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}^{\prime}={\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}\cap{\mathbf{L}}_{\alpha}$ and $A^{\prime}=A\cap{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$, and in addition ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}^{\prime},A^{\prime}\in{\mathfrak{M}}_{\alpha}$. By the elementarity, we have ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}^{\prime}={\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ and $A^{\prime}=A\cap{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}\in{\mathfrak{M}}_{\alpha}$, and $A^{\prime}$ is a maximal antichain, hence a pre-dense set, in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$. But then $A^{\prime}$ is a pre-dense set, hence, a maximal antichain, in the whole set ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ by Lemma 12.1(ii). Thus $A=A^{\prime}$ is countable. ∎

13 Generic model

This section presents some properties of $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic extensions ${\mathbf{L}}[G]$ of ${\mathbf{L}}$ obtained by adjoining a $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic set $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ to ${\mathbf{L}}$. Recall that the forcing notion ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}\in{\mathbf{L}}$ was introduced by Definition 11.5, along with some related notation.

The next lemma involves the coding system for continuous maps introduced in Section 7. If $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is generic over ${\mathbf{L}}$ and $\mathbf{c}\in\text{CCF}\cap{\mathbf{L}}$, then define $\mathbf{c}[G]=f_{\mathbf{c}}(\mathbf{a}[G])\in 2^{\omega}\cap{\mathbf{L}}[G]$; the definition of $\mathbf{a}[G]$ see 6.2.

Lemma 13.1** (continuous reading of names).**

If a set $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is generic over ${\mathbf{L}}$ and $x\in 2^{\omega}\cap{\mathbf{L}}[G]$ then there exists a code $\mathbf{c}\in\text{CCF}\cap{\mathbf{L}}$ such that $x=\mathbf{c}[G]$.

Proof**.**

One of basic forcing lemmas (Lemma 2.5 in [1, Chap. 4]) claims that there is a $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-name $t\in{\mathbf{L}}$ for $x$, satisfying $x=t[G]$ (the $\hskip 0.86108pt{G}$-valuation of $t$), and it can be assumed that ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ forces that $t$ is valuated as a real in $2^{\omega}$. Then the sets $F_{ni}=\{\hskip 0.43057pt{T\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptT\text{ forces }t(n)=i\hskip 0.43057pt\}$ ($n<\omega$ and $i=0,1$) satisfy the following:

1. (1)

the indexed set $\langle F_{ni}\rangle_{n<\omega\land i=0,1}$ belongs to ${\mathbf{L}}$; 2. (2)

if $n<\omega$, $S\in F_{n0}$, $T\in F_{n1}$, then $S,T$ are incompatible in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$; 3. (3)

if $n<\omega$ then the set $F_{n}=F_{n0}\cup F_{n1}$ is open dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$.

We argue in ${\mathbf{L}}$. Pick a maximal antichain $A_{n}\subseteq F_{n}$ in each $F_{n}$. Then all sets $A_{n}$ are maximal antichains in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ by (3), and all $A_{n}$ are countable by Corollary 12.4. Therefore there is an ordinal $\alpha<\omega_{1}^{\mathbf{L}}$ such that the set $\bigcup_{n}A_{n}\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ and the sequence $\langle A_{n}\rangle_{n<\omega}$ belong to ${\mathfrak{M}}_{\alpha}$. Note that $G\cap{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}\neq\varnothing$ by Lemma 12.1(iii); let $U\in{G\cap{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}}$. As ${{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}}\mathrel{\sqsubset_{{\mathfrak{M}}_{\alpha}}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ by Lemma 12.1(i), we have $U\subseteq^{\text{\tt fin}}\bigcup A_{n}$ for all $n$, there exists a finite set $A^{\prime}_{n}\subseteq A_{n}$ such that $U\subseteq\bigcup A^{\prime}_{n}$.

Let $A^{\prime}_{ni}=A^{\prime}_{n}\cap F_{ni}$ and $X_{ni}=[U]\cap\bigcup_{T\in A^{\prime}_{ni}}[T]$, $i=0,1$. We claim that $X_{n0}\cap X_{n1}=\varnothing$. Indeed otherwise there exist trees $T_{i}\in F_{ni}$ such that $Z=[U]\cap[T_{0}]\cap[T_{1}]$ is non-empty. By Lemma 8.2(i), $Z$ is clopen in $[U]$. Therefore there is a tree $U^{\prime}\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ such that $[U^{\prime}]\subseteq Z$, hence, $T_{0}$ and $T_{1}$ are compatible in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$, which contradicts (2) by construction. Thus indeed $X_{n0}\cap X_{n1}=\varnothing$.

As clearly $X_{n0}\cup X_{n1}=[U]$, we can define a continuous $g:[U]\to 2^{\omega}$ such that $g(x)(n)=i$ iff $x\in X_{ni}$. The map $g$ can be extended to a continuous $f:2^{\omega}\to 2^{\omega}$, that is $f(x)(n)=i$ whenever $x\in X_{ni}$ — for all $n<\omega$ and $i=0,1$. We have $f=f_{\mathbf{c}}$, where $\mathbf{c}\in\text{CCF}\cap{\mathbf{L}}$.

We argue in ${\mathbf{L}}[G]$. We skip a routine verification of $x=\mathbf{c}[G]$. ∎

Lemma 13.2**.**

If $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is generic over ${\mathbf{L}}$ then $\mathbf{a}[G]$ is not OD in ${\mathbf{L}}[G]$.

Proof**.**

Assume to the contrary that $\vartheta(x)$ is a formula with ordinal parameters, and a tree $T\in G$ $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-forces that $\mathbf{a}[G]$ is the only real $x\in 2^{\omega}$ satisfying $\vartheta(x)$. Let $s=\text{\tt stem}(T)$ and $n=\mathop{\tt lh}(s)$. Then $T$ contains both $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}0$ and $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}1$. Then either $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}0\subset\mathbf{a}[G]$ or $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}1\subset\mathbf{a}[G]$. Let say $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}0\subset\mathbf{a}[G]$.

Let $\sigma=0^{n}{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}1$, so that the strings $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}0$, $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}1$, $\sigma$ belong to $2^{n+1},$ $s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}1=\sigma\boldsymbol{\cdot}s{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}0$, and $\sigma\boldsymbol{\cdot}T=T$ by Lemma 4.3(ii). As ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is invariant under the action of $\sigma$, the set $G^{\prime}=\sigma\boldsymbol{\cdot}G=\{\hskip 0.43057pt{\sigma\boldsymbol{\cdot}S\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptS\in G\hskip 0.43057pt\}$ is $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic over ${\mathbf{L}}$, and $T=\sigma\boldsymbol{\cdot}T\in G^{\prime}$. We conctude that it is true in ${\mathbf{L}}[G^{\prime}]={\mathbf{L}}[G]$ that $\mathbf{a}[G^{\prime}]=\sigma\boldsymbol{\cdot}\mathbf{a}[G]$ is still the unique real in $2^{\omega}$ satisfying $\vartheta(\mathbf{a}[G^{\prime}])$. But $\mathbf{a}[G^{\prime}]\neq\mathbf{a}[G]$! ∎

14 Definability of the set of generic reals

We continue to argue in the context of Definition 11.5. The goal of this section is to study the definability of the set of all $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic reals $x\in 2^{\omega}$ in $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic extensions of ${\mathbf{L}}$.

Lemma 14.1**.**

In an transitive model of ZF extending ${\mathbf{L}}$, it is true that a real $x\in 2^{\omega}$ is $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic over ${\mathbf{L}}$ iff $x$ belongs to the set $\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}=\bigcap_{\alpha<\omega_{1}^{\mathbf{L}}}\bigcup_{T\in{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}_{\alpha}}[T]$.

Proof**.**

All sets ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ are pre-dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ by Lemma 12.1(iii). Therefore all $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic reals belong to $\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}$. On the other hand, any maximal antichain $A\in{\mathbf{L}}\hskip 0.43057pt,\linebreak[0]\>A\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is countable in ${\mathbf{L}}$ by Corollary 12.4, and hence $A\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ and $A\in{\mathfrak{M}}_{\alpha}$ for some index $\alpha<\omega_{1}^{\mathbf{L}}$. But then every tree $T\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ satisfies $T\subseteq^{\text{\tt fin}}\bigcup A$ by Lemma 8.2(ii). We conclude that $\bigcup_{T\in{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}_{\alpha}}[T]\subseteq\bigcup_{S\in A}[S]$. ∎

According to the next lemma, $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic extensions do not contain $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic reals, except the real $\mathbf{a}[G]$ itself and reals connected to $\mathbf{a}[G]$ in terms of the equivalence relation $\mathrel{{\text{\sf E}}_{0}}$ (see Footnote 5). We observe that the $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class

[TABLE]

of any real $x\in 2^{\omega}$ is a countable set.

Lemma 14.2**.**

Let $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ be a $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic set over ${\mathbf{L}}$. Then it is true in ${\mathbf{L}}[G]$ that $\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}=[\mathbf{a}[G]]_{\mathrel{{\text{\sf E}}_{0}}}$.

Proof**.**

The real $\mathbf{a}[G]$ is $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic, hence $\mathbf{a}[G]\in\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}$ by Lemma 14.1. Yet every set ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ is a ST-forcing, that is by definition it is closed under the action $s\boldsymbol{\cdot}T$ of any string $s\in 2^{<\omega}.$ This implies $[\mathbf{a}[G]]_{\mathrel{{\text{\sf E}}_{0}}}\subseteq\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}$.

To prove in the other direction, assume to the contrary that $x\in{\mathbf{L}}[G]\cap 2^{\omega}$, $x\in\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}\smallsetminus[\mathbf{a}[G]]_{\mathrel{{\text{\sf E}}_{0}}}$. By Lemma 13.1, there is a code $\mathbf{c}\in{\mathbf{L}}\cap\text{CCF}$ such that $x=\mathbf{c}[G]$. By the contrary assumption there is a tree $T_{0}\in G$ which forces

$\mathbf{c}[\underline{G}]\in\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}\smallsetminus[\mathbf{a}[\underline{G}]]_{\mathrel{{\text{\sf E}}_{0}}}$.

Fix an ordinal $\alpha<\omega_{1}^{\mathbf{L}}$ such that $\mathbf{c}\in{\mathfrak{M}}_{\alpha}$. We claim that (in ${\mathbf{L}}$) the function $f=f_{\mathbf{c}}$ is regular on $T_{0}$ inside ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$.

Indeed otherwise there exist $\sigma\in 2^{<\omega}$ and $T\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}$ such that $T\subseteq T_{0}$ and $\sigma\boldsymbol{\cdot}f(x)=x$ for all $x\in T$. Then $T$ forces $\sigma\boldsymbol{\cdot}\mathbf{c}[\underline{G}]=\mathbf{a}[\underline{G}]$, that is, forces $\mathbf{c}[\underline{G}]\in[\mathbf{a}[\underline{G}]]_{\mathrel{{\text{\sf E}}_{0}}}$, contrary to the choice of $T_{0}$. The regularity is established.

Recall that ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{<\alpha}\mathrel{\sqsubset_{{\mathfrak{M}}_{\alpha}}}{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ by Lemma 12.1(i). Therefore by 8.1(A), there is a tree $U\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$, $U\subseteq T_{0}$. And by 8.1(C) if $V\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ then $[V]\cap({f}\text{\rm\hskip 0.86108pt''}{[U]})=\varnothing$. Thus if $V\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}_{\alpha}$ then $U$ forces $\mathbf{c}[\underline{G}]\notin[V]$, hence forces $\mathbf{c}[\underline{G}]\notin\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}$, which contradicts to the choice of $T_{0}$. ∎

Corollary 14.3**.**

Let $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ be a $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic set over ${\mathbf{L}}$. Then the $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class $[\mathbf{a}[G]]_{\mathrel{{\text{\sf E}}_{0}}}$ is a $\varPi^{1}_{\mathbb{n}}$ set in ${\mathbf{L}}[G]$.

Proof**.**

A routine verification of $\text{\rm GEN}_{\hskip 0.3014pt{\mathbb{P}}\hskip 0.3014pt}\in\varPi^{1}_{\mathbb{n}}$ in ${\mathbf{L}}[G]$ using the property 11.4(i) of the sequence $\overline{\mathbf{P}}$ is left to the reader. ∎

15 Auxiliary forcing relation

Here we introduce a key tool for the proof of claim (i) of Theorem 1.1. This is a forcing-like relation $\mathop{\tt forc}$. It is not explicitly connected with the forcing notion ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ (but rather connected with the full Silver forcing $\mathbf{ST}$), however it will be compatible with ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ for formulas of certain quantifier complexity (Lemma 17.1). The crucial advantage of $\mathop{\tt forc}$ will be its invariance under two certain groups of transformations (Lemma 16.1), a property that cannot be expected for ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$. This will be the key argument in the proof of Theorem 17.2 below.

We argue in ${\mathbf{L}}$.

We consider a language $\mathscr{L}$ containing variables $i,j,k,\dots$ of type 0 with the domain $\omega$ and variables $x,y,z,\dots$ of type 1 with the domain $2^{\omega}$. Terms are variables of type 0 and expressions like $x(k)$. Atomic formulas are those of the form $R(t_{1},\dots,t_{n})$, where $R\subseteq\omega^{n}$ is any $\hskip 0.86108pt{n}$-ary relation on $\omega$ in ${\mathbf{L}}$. A formula is arithmetic if it does not contain variables of type 1. Formulas of the form

[TABLE]

where $\Psi$ is arithmetic, are of types $\mathscr{L}\varSigma^{1}_{n}$, resp., $\mathscr{L}\varPi^{1}_{n}$.

In addition we allow codes $\mathbf{c}\in\text{CCF}$ to substitute free variables of type 1. The semantics is as follows. Let $\varphi:=\varphi(\mathbf{c}_{1},\dots,\mathbf{c}_{k})$ be an $\hskip 0.86108pt{\mathscr{L}}$-formula, with all codes in CCF explicitly indicated, and let $x\in 2^{\omega}$. By $\varphi[x]$ we denote the formula $\varphi(f_{\mathbf{c}_{1}}(x),\dots,f_{\mathbf{c}_{k}}(x))$, where all $f_{\mathbf{c}_{i}}(x)$ are reals in $2^{\omega}$, of course.

Arithmetic formulas and those in $\mathscr{L}\varSigma^{1}_{n}\cup\mathscr{L}\varPi^{1}_{n}$, $n\geq 1$, will be called normal. If $\varphi$ is a formula in $\mathscr{L}\varSigma^{1}_{n}$ or $\mathscr{L}\varPi^{1}_{n}$ then $\varphi^{-}$ is the result of canonical transformation of $\neg\>\varphi$ to $\mathscr{L}\varPi^{1}_{n}$, resp., $\mathscr{L}\varSigma^{1}_{n}$ form. For arithmetic formulas, let $\varphi^{-}:=\neg\>\varphi$.

Definition 15.1** (in ${\mathbf{L}}$).**

We define a relation $T\mathop{\tt forc}\varphi$ between trees $T\in\mathbf{ST}$ and closed normal $\hskip 0.86108pt{\mathscr{L}}$-formulas:

1. (I)

if $\varphi$ is a closed $\hskip 0.86108pt{\mathscr{L}}$-formula, arithmetic or in $\mathscr{L}\varSigma^{1}_{1}\cup\mathscr{L}\varPi^{1}_{1}$, then $T\mathop{\tt forc}\varphi$ iff $\varphi[x]$ holds for all $x\in[T]$; 2. (II)

if $\varphi:={\exists\,}x\,\psi(x)$ is a closed $\mathscr{L}\varSigma^{1}_{n+1}$ formula, $n\geq 1$ ($\psi$ being of type $\mathscr{L}\varPi^{1}_{n}$), then $T\mathop{\tt forc}\varphi$ iff there is a code $\mathbf{c}\in\text{CCF}$ such that $T\mathop{\tt forc}\psi(\mathbf{c})$; 3. (III)

if $\varphi$ is a closed $\mathscr{L}\varPi^{1}_{n}$ formula, $n\geq 2$, then $T\mathop{\tt forc}\varphi$ iff there is no tree $S\in\mathbf{ST}$ such that $S\subseteq T$ and $S\mathop{\tt forc}\psi^{-}$.

Let $\text{Forc}(\varphi)=\{\hskip 0.43057pt{T\in\mathbf{ST}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptT\mathop{\tt forc}\varphi\hskip 0.43057pt\}$ and $\text{Des}(\varphi)=\text{Forc}(\varphi)\cup\text{Forc}(\varphi^{-})$. ∎

Lemma 15.2** (in ${\mathbf{L}}$).**

If $m\geq 2$ and $\varphi$ is a closed formula in $\mathscr{L}\varSigma^{1}_{m}$, resp., $\mathscr{L}\varPi^{1}_{m}$, then the set $\text{Forc}(\varphi)$ belongs to $\varSigma^{\mathrm{HC}}_{m-1}(\mathrm{HC})$, resp., $\varPi^{\mathrm{HC}}_{m-1}(\mathrm{HC})$.

Proof**.**

For $\mathscr{L}\varPi^{1}_{1}$ formulas, Definition 15.1(I) implies $\text{Forc}(\varphi)\in{\mathbf{\Pi}}^{1}_{1}$, thus $\text{Forc}(\varphi)$ belongs to $\varDelta^{\mathrm{HC}}_{1}(\mathrm{HC})$. Then argue by induction on Definition 15.1(II),(III). ∎

Recall that a number $\mathbb{n}\geq 3$ is fixed by Definition 11.5.

Lemma 15.3** (in ${\mathbf{L}}$).**

Let $\varphi$ be a closed normal $\hskip 0.86108pt{\mathscr{L}}$-formula. Then the set $\text{Des}(\varphi)$ is dense in $\mathbf{ST}$. If $\varphi$ is of type $\mathscr{L}\varSigma^{1}_{m}$, $m<\mathbb{n}$, then $\text{Des}(\varphi)\cap{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is dense in ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$.

Proof**.**

It suffices to establish the density of $\text{Des}(\varphi)$ for formulas $\varphi$ as in (I). If $\varphi$ is such then the set $X(\varphi)=\{\hskip 0.43057pt{x\in T\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\varphi[x]\hskip 0.43057pt\}$ belongs to ${\mathbf{\Sigma}}^{1}_{1}\cup{\mathbf{\Pi}}^{1}_{1}$, that is, it has the Baire property inside $[T]$. Therefore at least one of the two complimentary sets $X(\varphi)$, $X(\varphi^{-})$ is not meager in $[T]$. It remains to apply Corollary 5.4.

The second claim follows from the first one by lemmas 15.2 and 12.1(iv). ∎

16 Invariance

It happens that the relation $\mathop{\tt forc}$ is invariant under two rather natural groups of transformations of $\mathbf{ST}$. Here we prove the invariance. We still argue in ${\mathbf{L}}$.

First group. Let $h\subseteq\omega$. If $x\in 2^{\omega}$ then a real $h\boldsymbol{\cdot}x\in 2^{\omega}$ is defined by $(h\boldsymbol{\cdot}x)(j)=1-x(j)$ for $j\in h$, but $(h\boldsymbol{\cdot}x)(j)=x(j)$ for $j\notin h$. If $X\subseteq 2^{\omega}$ then let $h\boldsymbol{\cdot}X=\{\hskip 0.43057pt{h\boldsymbol{\cdot}x\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptx\in X\hskip 0.43057pt\}$. Accordingly, if $s\in 2^{<\omega}$ and $n=\mathop{\tt lh}(s)$ then a string $h\boldsymbol{\cdot}s\in 2^{<\omega}$ is defined by $\mathop{\tt dom}(h\boldsymbol{\cdot}s)=n=\mathop{\tt dom}s$ and if $j<n$ then $(h\boldsymbol{\cdot}x)(j)=1-x(j)$ for $j\in h$, but $(h\boldsymbol{\cdot}x)(j)=x(j)$ for $j\notin h$. If $T\subseteq 2^{<\omega}$ then let $h\boldsymbol{\cdot}T=\{\hskip 0.43057pt{h\boldsymbol{\cdot}s\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pts\in T\hskip 0.43057pt\}$. Then obviously $T\in\mathbf{ST}$ iff $h\boldsymbol{\cdot}T\in\mathbf{ST}$.

If $f:2^{\omega}\to 2^{\omega}$ then a function $h\boldsymbol{\cdot}f:2^{\omega}\to 2^{\omega}$ is defined by $(h\boldsymbol{\cdot}f)(x)=f(h\boldsymbol{\cdot}x)$, equivalently, $(h\boldsymbol{\cdot}f)(h\boldsymbol{\cdot}x)=f(x)$. If $f$ is continuous, then $f=f_{\mathbf{c}}$, where $\mathbf{c}\in\text{CCF}$, and there is a canonical definition of a code $h\boldsymbol{\cdot}\mathbf{c}\in\text{CCF}$ such that $h\boldsymbol{\cdot}(f_{\mathbf{c}})=f_{h\boldsymbol{\cdot}\mathbf{c}}$.

Finally if $\varphi:=\varphi(\mathbf{c}_{1},\dots,\mathbf{c}_{k})$ is a $\hskip 0.86108pt{\mathscr{L}}$-formula then let $h\varphi$ be the formula $\varphi(h\boldsymbol{\cdot}\mathbf{c}_{1},\dots,h\boldsymbol{\cdot}\mathbf{c}_{k})$. Then $(h\varphi)[h\boldsymbol{\cdot}x]$ coincides with $\varphi[x]$.

Second group. Let IB be the set of all idempotent bijections $b:\omega\overset{{\text{onto}}}{\longrightarrow}\omega$, that is, we require that $b(j)=b^{-1}(j)$, ${\forall\,}j$. If $x\in 2^{\omega}$ then define $b\boldsymbol{\cdot}x\in 2^{\omega}$ by $(b\boldsymbol{\cdot}x)(j)=x(b(j))$, ${\forall\,}j$. Let $b\boldsymbol{\cdot}X=\{\hskip 0.43057pt{b\boldsymbol{\cdot}x\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptx\in X\hskip 0.43057pt\}$, for $X\subseteq 2^{\omega}$. If $T\in\mathbf{ST}$ then put $b\boldsymbol{\cdot}T=\{\hskip 0.43057pt{x{\hskip 0.43057pt\restriction\hskip 1.29167pt}m\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptx\in(b\boldsymbol{\cdot}[T])\land m<\omega\hskip 0.43057pt\}$. Clearly $T\in\mathbf{ST}$ iff $b\boldsymbol{\cdot}T\in\mathbf{ST}$.

If $f:2^{\omega}\to 2^{\omega}$ then a function $b\boldsymbol{\cdot}f:2^{\omega}\to 2^{\omega}$ is defined similarly to the above by $(b\boldsymbol{\cdot}f)(x)=f(b\boldsymbol{\cdot}x)$, equivalently, $(b\boldsymbol{\cdot}f)(b\boldsymbol{\cdot}x)=f(x)$. If $f$ is continuous, then $f=f_{\mathbf{c}}$, where $\mathbf{c}\in\text{CCF}$, and still there is a canonical definition of a code $b\boldsymbol{\cdot}\mathbf{c}\in\text{CCF}$ such that $b\boldsymbol{\cdot}(f_{\mathbf{c}})=f_{b\boldsymbol{\cdot}\mathbf{c}}$.

And finally if $\varphi:=\varphi(\mathbf{c}_{1},\dots,\mathbf{c}_{k})$ is a $\hskip 0.86108pt{\mathscr{L}}$-formula then let $b\varphi$ be the formula $\varphi(b\boldsymbol{\cdot}\mathbf{c}_{1},\dots,b\boldsymbol{\cdot}\mathbf{c}_{k})$. Then $(b\varphi)[b\boldsymbol{\cdot}x]$ coincides with $\varphi[x]$.

Lemma 16.1** (in ${\mathbf{L}}$).**

Let $T\in\mathbf{ST}$ and $\varphi$ be a closed normal $\hskip 0.86108pt{\mathscr{L}}$-formula. Then

1. (i)

if $h\subseteq\omega$ then $T\mathop{\tt forc}\varphi$ iff ${(h\boldsymbol{\cdot}T)}\mathop{\tt forc}h\varphi$;** 2. (ii)

if $b\in\text{\rm IB}$ then $T\mathop{\tt forc}\varphi$ iff ${(b\boldsymbol{\cdot}T)}\mathop{\tt forc}b\varphi$.

Proof**.**

(i) If $\varphi$ is of type 15.1(I) then it suffices to note that, first, $[h\boldsymbol{\cdot}T]=\{\hskip 0.43057pt{h\boldsymbol{\cdot}x\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277ptx\in[T]\hskip 0.43057pt\}$, and second, if $x\in[T]$ then $\varphi[x]$ coincides with $(h\varphi)[h\boldsymbol{\cdot}x]$. A routine induction based on Definition 15.1(II),(III) is left to the reader. ∎

Corollary 16.2**.**

Let $\varphi$ be a closed normal $\hskip 0.86108pt{\mathscr{L}}$-formula, such that if a code $\mathbf{c}$ occurs in $\varphi$ then $f_{\mathbf{c}}$ is a constant. Assume that $S,T\in\mathbf{ST}$ and the splitting sets $\text{\tt spl}(S)$, $\text{\tt spl}(T)$ (see Section 4) are both co-infinite. Then $S\mathop{\tt forc}\varphi$ iff $T\mathop{\tt forc}\varphi$.

Proof**.**

Assume that $S\mathop{\tt forc}\varphi$. As both $\text{\tt spl}(S)$ and $\text{\tt spl}(T)$ are co-infinite (and they are infinite anyway), there is a bijection $b\in\text{\rm IB}$ such that ${b}\text{\rm\hskip 0.86108pt''}{(\text{\tt spl}(S))}=\text{\tt spl}(T)$. Then the tree $S^{\prime}=b\boldsymbol{\cdot}S$ satisfies $\text{\tt spl}(S^{\prime})=\text{\tt spl}(T)$, and still $S^{\prime}\mathop{\tt forc}\varphi$ by Lemma 16.1(ii), as all occurring codes define constant functions, and hence $b\varphi$ and $\varphi$ essentially coincide. Now, as $\text{\tt spl}(S^{\prime})=\text{\tt spl}(T)$, there is a set $h\subseteq\omega\smallsetminus\text{\tt spl}(S^{\prime})=\omega\smallsetminus\text{\tt spl}(T)$ such that $T=h\boldsymbol{\cdot}S^{\prime}$. Then $T\mathop{\tt forc}\varphi$ by Lemma 16.1(i) and the constantness of the codes involved, as above. ∎

17 The final argument

Recall that $\mathbb{n}\geq 3$ is fixed by Definition 11.5.

The last part of the proof of Theorem 1.1 will be Lemma 17.2. Note the key ingredient of the proof: we surprisingly approximate the forcing ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$, definitely non-invariant under the transformations considered in Section 16, by the invariant relation $\mathop{\tt forc}$, using the next lemma.

Lemma 17.1**.**

Assume that $1\leq n<\mathbb{n}$, $\varphi\in{\mathbf{L}}$ is a closed formula in $\mathscr{L}\varPi^{1}_{n}\cup\mathscr{L}\varSigma^{1}_{n+1}$, and a set $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is generic over ${\mathbf{L}}$. Then the sentence $\varphi[\mathbf{a}[G]]$ is true in ${\mathbf{L}}[G]$ if and only if ${\exists\,}T\in G\,(T\mathop{\tt forc}\varphi)$.

Proof**.**

Base of induction: $\varphi$ is arithmetic or belongs to $\mathscr{L}\varSigma^{1}_{1}\cup\mathscr{L}\varPi^{1}_{1}$, as in 15.1(I). If $T\in G$ and $T\mathop{\tt forc}\varphi$, then $\varphi[\mathbf{a}[G]]$ holds by the Shoenfield absoluteness theorem, as $\mathbf{a}[G]\in[T]$. The inverse holds by Lemma 15.3.

Step $\mathscr{L}\varPi^{1}_{n}\mathbin{\,\Longrightarrow\,}\mathscr{L}\varSigma^{1}_{n+1}$. Let $\varphi$ be ${\exists\,}x\,\psi(x)$ where $\psi$ is of type $\mathscr{L}\varPi^{1}_{n}$. Assume that $T\in G$ and $T\mathop{\tt forc}\varphi$. Then by Definition 15.1(II) there is a code $\mathbf{c}\in\text{CCF}\cap{\mathbf{L}}$ such that $T\mathop{\tt forc}\psi(\mathbf{c})$. By the inductive hypothesis, the formula $\psi(\mathbf{c})[\mathbf{a}[G]]$, that is, $\psi[G](f_{\mathbf{c}}(\mathbf{a}[G]))$, is true in ${\mathbf{L}}[G]$. But then $\varphi[\mathbf{a}[G]]$ is obviously true as well.

Conversely assume that $\varphi[\mathbf{a}[G]]$ is true. Then there is a real $y\in{\mathbf{L}}[G]\cap 2^{\omega}$ such that $\psi[G](y)$ is true. By Lemma 13.1, $y=f_{\mathbf{c}}(\mathbf{a}[G])$ for a code $\mathbf{c}\in\text{CCF}\cap{\mathbf{L}}$. But then $\psi(\mathbf{c})[\mathbf{a}[G]]$ is true in ${\mathbf{L}}[G]$. By the inductive hypothesis, there is a tree $T\in G$ satisfying $T\mathop{\tt forc}\psi(\mathbf{c})$. Then $T\mathop{\tt forc}\varphi$ as well.

Step $\mathscr{L}\varSigma^{1}_{n}\mathbin{\,\Longrightarrow\,}\mathscr{L}\varPi^{1}_{n}$. Let $\varphi$ be a $\mathscr{L}\varPi^{1}_{n}$ formula, $n\geq 2$. By Lemma 15.3, there is a tree $T\in G$ such that either $T\mathop{\tt forc}\varphi$ or $T\mathop{\tt forc}\varphi^{-}$. If $T\mathop{\tt forc}\varphi^{-}$ then $\varphi^{-}[\mathbf{a}[G]]$ is true by the inductive hypothesis, hence $\varphi[\mathbf{a}[G]]$ is false. Now assume that $T\mathop{\tt forc}\varphi$. We have to prove that $\varphi[\mathbf{a}[G]]$ is true. Suppose otherwise. Then $\varphi^{-}[\mathbf{a}[G]]$ is true. By the inductive hypothesis, there is a tree $S\in G$ such that $S\mathop{\tt forc}\varphi^{-}$. But the trees $S,T$ belong to the same generic set $G$, hence they are compatible, which leads to a contradiction with the assumption $T\mathop{\tt forc}\varphi$, according to Definition 15.1(III). ∎

Lemma 17.2**.**

If a set $G\subseteq{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ is $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic over ${\mathbf{L}}$, then it is true in ${\mathbf{L}}[G]$, that every countable $\varSigma^{1}_{\mathbb{n}}$ set $X\subseteq 2^{\omega}$ satisfies $Y\in{\mathbf{L}}$.

Proof**.**

We work in the context of Definition 11.5. The first part of the proof is to show that $Y\subseteq{\mathbf{L}}$. Suppose to the contrary that it holds in ${\mathbf{L}}[G]$ that $Y\subseteq 2^{\omega}$ is a countable $\varSigma^{1}_{\mathbb{n}}$ set, but $Y\not\subseteq{\mathbf{L}}$. We have $Y=\{\hskip 0.43057pt{y\in 2^{\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\varphi(y)\hskip 0.43057pt\}$ where $\varphi(y):={\exists\,}z\,\psi(y,z)$ is a $\varSigma^{1}_{\mathbb{n}}$ formula, that is, $\mathscr{L}\varSigma^{1}_{\mathbb{n}}$ formula without codes in CCF. There is a tree $T_{0}\in G$ which $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-forces that “$\{\hskip 0.43057pt{y\in 2^{\omega}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\varphi(y)\hskip 0.43057pt\}$ is a countable set and ${\exists\,}y(\varphi(y)\land y\notin{\mathbf{L}})$”. Our goal is to derive a contradiction.

By Lemma 13.1, there exist codes $\mathbf{c},\mathbf{d}\in\text{CCF}\cap{\mathbf{L}}$ such that the real $y_{0}=\mathbf{c}[G]=f_{\mathbf{c}}(\mathbf{a}[G])$ belongs to $Y$, so that $y_{0}\notin{\mathbf{L}}$ and $\varphi(\mathbf{c})[\mathbf{a}[G]]$ holds, that is, ${\exists\,}z\,\psi(\mathbf{c},z)[\mathbf{a}[G]]$, and finally $\mathbf{d}$ witnesses the existence quantifier, so that the sentence $\psi(\mathbf{c},\mathbf{d})[\mathbf{a}[G]]$ holds in ${\mathbf{L}}[G]$. By Lemma 17.1 as $\psi$ is a $\mathscr{L}\varPi^{1}_{\mathbb{n}-1}$ formula, there is a tree $T_{1}\in G$ satisfying $T_{1}\mathop{\tt forc}\psi(\mathbf{c},\mathbf{d})$.

We can wlog assume that $T_{1}\subseteq T_{0}$ and that, in ${\mathbf{L}}$, the map $f_{\mathbf{c}}$ is either a constant or a bijection on $[T_{1}]$, by Lemma 12.1(v).

Case 1: $f_{\mathbf{c}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}[T_{1}]$ is a constant, that is, there exists a real $y_{1}\in 2^{\omega}\cap{\mathbf{L}}$ such that $f_{\mathbf{c}}(x)=y_{1}$ for all $x\in[T_{1}]$. But then $y_{1}=f_{\mathbf{c}}(\mathbf{a}[G])=y_{0}$, however $y_{0}\notin{\mathbf{L}}$ while $y_{1}\in{\mathbf{L}}$, which is a contradiction.

Case 2: $f_{\mathbf{c}}{\hskip 0.43057pt\restriction\hskip 1.29167pt}[T_{1}]$ is a bijection. As $T_{1}$ is a Silver tree, the set $H=\text{\tt spl}(T_{1})\subseteq\omega$ of all its splitting levels is infinite (Definition 4.1). Let $h\in{\mathbf{L}}$, $h\subseteq H$. Then $h\boldsymbol{\cdot}T_{1}=T_{1}$, and we have $T_{1}\mathop{\tt forc}\psi(h\boldsymbol{\cdot}\mathbf{c},h\boldsymbol{\cdot}\mathbf{d})$ By Lemma 16.1. Therefore the formula $\psi(h\boldsymbol{\cdot}\mathbf{c},h\boldsymbol{\cdot}\mathbf{d})[\mathbf{a}[G]]$ is true in ${\mathbf{L}}[G]$ by Lemma 17.1. But this formula coincides with $\psi(f_{h\boldsymbol{\cdot}\mathbf{c}}(\mathbf{a}[G]),f_{h\boldsymbol{\cdot}\mathbf{d}}(\mathbf{a}[D]))$, hence we have $\varphi(f_{h\boldsymbol{\cdot}\mathbf{c}}(\mathbf{a}[G]))$ in ${\mathbf{L}}[G]$. This implies $f_{h\boldsymbol{\cdot}\mathbf{c}}(\mathbf{a}[G])\in Y$, or equivalently, $f_{\mathbf{c}}(h\boldsymbol{\cdot}\mathbf{a}[G])\in Y$.

However if sets $h,h^{\prime}\in{\mathbf{L}}$, $h\cup h^{\prime}\subseteq H$, satisfy $h\neq h^{\prime}$ then $h\boldsymbol{\cdot}\mathbf{a}[G]\neq h^{\prime}\boldsymbol{\cdot}\mathbf{a}[G]$, and hence $f_{\mathbf{c}}(h\boldsymbol{\cdot}\mathbf{a}[G])\neq f_{\mathbf{c}}(h^{\prime}\boldsymbol{\cdot}\mathbf{a}[G])$, as $f_{\mathbf{c}}$ is a bijection on $[T_{1}]$ (and $\mathbf{a}[G]\in[T_{1}]$). Thus picking different sets $h\in{\mathbf{L}}\cap{\hskip 0.43057pt{\mathscr{P}}\hskip 0.43057pt}(H)$ we get uncountably many different elements of the set $Y$ in ${\mathbf{L}}[G]$, which contradicts to the choice of $Y$.

The proof of $Y\subseteq{\mathbf{L}}$ is accomplished.

To prove $Y\in{\mathbf{L}}$, a stronger statement, it suffices now to show that if $y_{0}\in 2^{\omega}\cap{\mathbf{L}}$ then $y_{0}\in Y$ iff ${\exists\,}T\in\mathbf{ST}\,(T\mathop{\tt forc}\varphi(\mathbf{c}_{0}))$, where $\mathbf{c}_{0}\in\text{CCF}\cap{\mathbf{L}}$ is the code of the constant function $f_{\mathbf{c}_{0}}(x)=y_{0}$, ${\forall\,}x\in 2^{\omega}$.

If $y_{0}\in Y$ then the formula $\varphi(y_{0})$, equal to $\varphi(\mathbf{c}_{0})[x]$ for any $x$, is true in ${\mathbf{L}}[G]$ by the choice of $\varphi$. It follows by Lemma 17.1 that there is a tree $T\in G$ satisfying $T\mathop{\tt forc}\varphi(\mathbf{c}_{0})$, as required.

Now suppose that $T\in\mathbf{ST}$ (not necessarily $\in{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$!) and $T\mathop{\tt forc}\varphi(\mathbf{c}_{0})$. As the set $D=\{\hskip 0.43057pt{T\in\mathbf{ST}\hskip 2.15277pt{:}}\linebreak[0]\hskip 2.15277pt\text{\tt spl}(T)\text{ is co-infinite}\hskip 0.43057pt\}$ (see Lemma 12.1(vi)) is open dense in $\mathbf{ST}$, we can assume that $\text{\tt spl}(T)$ is co-infinite. On the other hand, it follows from Lemma 12.1(vi) that there is a tree $S\in G\cap D$, so that $\text{\tt spl}(S)$ is co-infinite as well. Now we have $S\mathop{\tt forc}\varphi(\mathbf{c}_{0})$ by Corollary 16.2, and then $\varphi(\mathbf{c}_{0})[\mathbf{a}[G]]$ is true in ${\mathbf{L}}[G]$ by Lemma 17.1, that is, $\varphi(y_{0})$ holds in ${\mathbf{L}}[G]$, and $y_{0}\in Y$, as required. ∎

Proof** **(Theorem 1.1, the main theorem).

We assert that any $\hskip 0.86108pt{{\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}}$-generic extension ${\mathbf{L}}[G]={\mathbf{L}}[\mathbf{a}[G]]$ satisfies conditions (i), (ii), (iii) of the theorem. That $\mathbf{a}[G]\notin\text{\rm OD}$ in (i) follows by Lemma 13.2. The minimality follows from Lemma 12.1(v) by Lemma 13.1 (continuous reading of names). We further have (ii) by Corollary 14.3, and we have (iii) by Lemma 17.2. ∎

Index

• action

• $b\boldsymbol{\cdot}\mathbf{c}$ §16

• $b\varphi$ §16

• $h\boldsymbol{\cdot}\mathbf{c}$ §16

• $h\boldsymbol{\cdot}s$ §16

• $h\boldsymbol{\cdot}T$ §16

• $h\boldsymbol{\cdot}X$ §16

• $h\boldsymbol{\cdot}x$ §16

• $h\varphi$ §16

• $s\boldsymbol{\cdot}T$ §4

• $s\boldsymbol{\cdot}t$ §4

• $s\boldsymbol{\cdot}X$ §4

• $s\boldsymbol{\cdot}x$ §4

• blocks Definition 11.3

• negatively Definition 11.3

• positively Definition 11.3

• code, ${\text{code}}(f)$ §7

• collage Definition 6.5

• definability classes

• $\varSigma^{\mathrm{HC}}_{n},\,\varPi^{\mathrm{HC}}_{n},\,\varDelta^{\mathrm{HC}}_{n}$ §11

• $\varSigma^{\mathrm{HC}}_{n}(\mathrm{HC}),\,\varPi^{\mathrm{HC}}_{n}(\mathrm{HC})$ item (ii)

• equivalence relation

• $[x]_{\mathrel{{\text{\sf E}}_{0}}}$, $\hskip 0.86108pt{\mathrel{{\text{\sf E}}_{0}}}$-class §14

• $\mathrel{{\text{\sf E}}_{0}}$ §1

• Vitali, $\mathrel{\text{\sf VIT}}$ §2

• forcing

• by Silver trees Definition 6.1

• Cohen forcing, $\mathbf{P}_{\text{\rm coh}}$ Example 6.3

• extension, $\mathrel{\sqsubset}$ §8

• Jensen’s forcing, $\boldsymbol{J}$ §1

• invariant, $\boldsymbol{J}^{\text{inv}}$ §1

• $\hskip 0.86108pt{{\mathfrak{M}}}$-extension, $\mathrel{\sqsubset_{{\mathfrak{M}}}}$ Definition 8.1

• Silver forcing, $\mathbf{ST}$ Example 6.3

• ST-forcing Definition 6.1

• $\mathbf{STF}$ §11

• ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ Definition 11.5

• formula

• arithmetic §15

• language $\mathscr{L}$ §15

• $\mathscr{L}\varSigma^{1}_{n}$, $\mathscr{L}\varPi^{1}_{n}$ §15

• normal §15

• $\varphi^{-}$ §15

• $\varphi[x]$ §15

• function

• code, ${\text{code}}(f)$ §7

• regular Definition 7.1

• generic real

• $\mathbf{a}[G]$ Remark 6.2

• hereditarily countable

• $\mathrm{HC}$ §11

• map

• regular Definition 7.1

• $\hskip 0.86108pt{{\mathfrak{M}}}$-extension, $\mathrel{\sqsubset_{{\mathfrak{M}}}}$ Definition 8.1

• model

• ${\mathfrak{M}}_{\alpha}$ Definition 11.5

• ${\mathfrak{M}}(\overline{\mathbf{P}})$ §11

• number $\mathbb{n}\geq 3$ Definition 11.5

• regular Definition 7.1

• splitting §5

• $T{({\to\hskip 0.86108pt}i)}$ §5

• $T{({\to\hskip 0.86108pt}s)}$ §5

• stem, $\text{\tt stem}(T)$ Definition 4.1

• string §4

• $2^{<\omega},$ dyadic strings §4

• $2^{n},$ strings of length $n$ §4

• extension, $\subseteq$, $\subset$ §4

• length §4

• $u_{k}(T)$ Definition 4.1

• $\Lambda$, the empty string §4

• support

• $|\varphi|$ Definition 9.1

• system Definition 9.1

• $|\varphi|$ Definition 9.1

• extension, $\psi\preccurlyeq\varphi$ §9

• $\text{Sys}(\mathbf{P})$ Definition 9.1

• $T^{\varphi}_{\xi m}(n)$ Definition 9.1

• $T^{\varphi}_{\xi m}(s)$ Definition 9.1

• $\nu^{\varphi}_{\xi m}$ Definition 9.1

• $\tau^{\varphi}_{\xi m}$ Definition 9.1

• theory

• $\text{ZFC}^{\prime}$ §8

• tree

• branch Definition 4.1

• restricted $T{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}$ Definition 4.1

• Silver, $\mathbf{ST}$ Definition 4.1

• stem, $\text{\tt stem}(T)$ Definition 4.1

• $B[s]$ Example 4.2

• ${\mathbf{T}}^{\mathbf{\Phi}}_{\xi m}(n)$ Definition 9.3

• $T^{\varphi}_{\xi m}(n)$ Definition 9.1

• ${\mathbf{T}}^{\mathbf{\Phi}}_{\xi m}(s)$ Definition 9.3

• $T^{\varphi}_{\xi m}(s)$ Definition 9.1

• ${\mathbf{U}}^{\mathbf{\Phi}}_{\xi m}$ Definition 9.3

• $2^{<\omega}$ §4

• $\mathbf{a}[G]$, generic real Remark 6.2

• $B[s]$ Example 4.2

• ${\text{code}}(f)$ §7

• $\text{Colg}_{n}(\mathbf{P})$ Definition 6.5

• $\mathrel{{\text{\sf E}}_{0}}$ §1

• $\varphi^{-}$ §15

• $|\varphi|$ Definition 9.1

• $\varphi[x]$ §15

• $\mathrm{HC}$ §11

• $\boldsymbol{J}$ §1

• $\boldsymbol{J}^{\text{inv}}$ §1

• $\Lambda$, the empty string §4

• $\mathop{\tt lh}(s)$, length §4

• $\mathscr{L}\varPi^{1}_{n}$ §15

• $\mathscr{L}\varSigma^{1}_{n}$ §15

• ${\mathfrak{M}}_{\alpha}$ Definition 11.5

• ${\mathfrak{M}}(\overline{\mathbf{P}})$ §11

• $\text{Sys}(\mathbf{P})$ Definition 9.1

• $\mathbb{n}\geq 3$ Definition 11.5

• $\nu^{\varphi}_{\xi m}$ Definition 9.1

• $\mathbf{P}_{\text{\rm coh}}$ Example 6.3

• ${\hskip 0.43057pt{\mathbb{P}}\hskip 0.43057pt}$ Definition 11.5

• $s[u]$ Proof

• $\varSigma^{\mathrm{HC}}_{n},\,\varPi^{\mathrm{HC}}_{n},\,\varDelta^{\mathrm{HC}}_{n}$ §11

• $\varSigma^{\mathrm{HC}}_{n}(\mathrm{HC}),\,\varPi^{\mathrm{HC}}_{n}(\mathrm{HC})$ item (ii)

• $\text{\tt spl}_{n}(T)$ Definition 4.1

• $\text{\tt spl}(T)$ Definition 4.1

• $\subseteq^{\text{\tt fin}}$ item (B)

• $\mathrel{\sqsubset}$ Definition 8.1

• $\mathrel{\sqsubset_{{\mathfrak{M}}}}$ Definition 8.1

• $S\mathrel{{\subseteq}_{n}}T$ §5

• $\text{\tt stem}(T)$ Definition 4.1

• $\mathbf{STF}$ §11

• $s\subseteq t$ §4

• $s\subset t$ §4

• $\mathrel{{\subseteq}_{n}}$ §5

• $T{({\to\hskip 0.86108pt}i)}$ §5

• $T{({\to\hskip 0.86108pt}s)}$ §5

• $\tau^{\varphi}_{\xi m}$ Definition 9.1

• ${\mathbf{T}}^{\mathbf{\Phi}}_{\xi m}(n)$ Definition 9.3

• $T^{\varphi}_{\xi m}(n)$ Definition 9.1

• ${\mathbf{T}}^{\mathbf{\Phi}}_{\xi m}(s)$ Definition 9.3

• $T^{\varphi}_{\xi m}(s)$ Definition 9.1

• $[T]$ Definition 4.1

• $T{\hskip 0.43057pt\restriction\hskip 1.29167pt}_{\hskip 0.1507ptu}$ Definition 4.1

• $t{\mathbin{\hskip 0.86108pt{}^{\smallfrown}}}i$ §4

• $u[s]$ Proof

• ${\mathbf{U}}^{\mathbf{\Phi}}_{\xi m}$ Definition 9.3

• $\mathrel{\text{\sf VIT}}$ §2

• $[x]_{\mathrel{{\text{\sf E}}_{0}}}$ §14

• $\text{ZFC}^{\prime}$ §8