The Special Aronszajn Tree Property at $\kappa^+$ and $\square_{\kappa, 2}$

TL;DR

This paper proves the consistency of the square principle 2 with all 2-Aronszajn trees being special for any regular kappa, extending results to all regular cardinals using advanced set-theoretic methods.

Contribution

It establishes the consistency of 2 with the special Aronszajn tree property at 22 for all regular cardinals, a significant extension in set theory.

Findings

2 is consistent with all 2-Aronszajn trees being special.

The result applies simultaneously to all regular 2.

It shows 2 cannot generally be strengthened to 2.

Abstract

We show that for any regular cardinal $\kappa$, $\square_{\kappa, 2}$ is consistent with "all $\kappa^+$-Aronszajn trees are special." By a result of Shelah and Stanley this is optimal in the sense that $\square_{\kappa, 2}$ may not be strengthened to $\square_{\kappa}$. Using methods of Golshani and Hayut we obtain our consistency result simultaneously for all regular $\kappa$.

Equations

$$D=\bigcup _{1\le \zeta <\xi }\mathrm{dom}(p_{\xi })$$

$$\displaystyle C=\bigcup_{\zeta<\xi}{C_{\zeta}}\cup\bigg{\{}\sup{\bigcup_{\zeta<\xi}{C_{\zeta}}}\bigg{\}}$$

$$p_{\xi }(\alpha )=\left\{\max (\mathrm{dom}(p_{\zeta })):1\le \zeta <\xi \right\}$$

$$\displaystyle p^{*}\big{(}\sup_{\xi<\nu}{(\max{(\operatorname{dom}{p_{\xi}})})}\big{)}=\left\{t^{*}\right\}$$

$$t^{\ast }=\bigcup _{\xi <\nu }{t}_{\xi }$$

$${⊩}_{{\mathbb{I}}^1}^{V^{{\mathbb{I}}^2}}\check{s}{\perp }_{\dot{T}}\check{t}$$

$${\vec{\mathbb{I}}}^2$$

$${\dot{\vec{\mathbb{I}}}}^1$$

$${\vec{\mathbb{I}}}^2$$

$${\dot{\vec{\mathbb{I}}}}^1$$

$$\kappa ={\mu }^{+}\wedge {\square }_{\mu ,2}\wedge \mathsf{SATP}(\kappa )\wedge 2^{\mu }\ge \delta$$

$$\left\{\alpha <\kappa :\left(V_{\alpha },\in ,R\cap V_{\alpha }\right)\models \psi \right\}\in \mathcal{F}$$

$$⟨r_{\eta },{\xi }_{\eta },{\check{x}}_{\eta }^L,{\check{x}}_{\eta }^R:\eta <\vartheta ⟩$$

$${⊩}_{[({\mathbb{I}}_{\overline{\stackrel{ˉ}{\gamma }}}^2\ast {\dot{\mathbb{I}}}_{\overline{\stackrel{ˉ}{\gamma }}}^1)\cap M]/(H\ast J_i)}\left(\exists \xi <{\mu }_0^i\right)\left({\check{b}}_{\xi }^i={\dot{b}}^i\right)$$

$$b\in V[H][J_L]\cap V[H][J_R]$$

$$(p_0^L\upharpoonright \overline{\stackrel{ˉ}{\gamma }},r_0\upharpoonright \overline{\stackrel{ˉ}{\gamma }})$$

$$(p_0^R\upharpoonright \overline{\stackrel{ˉ}{\gamma }},r_0\upharpoonright \overline{\stackrel{ˉ}{\gamma }})$$

$$\left(p_0^L\upharpoonright M_{\alpha }\right)\left(0\right)\ast {\check{t}}_0^L,$$

$$\left(p_0^R\upharpoonright M_{\alpha }\right)\left(0\right)\ast {\check{t}}_0^R$$

$$t_{\nu }^L$$

$$t_{\nu }^R$$

$${\bar{q}}_{\nu }{⊩}_{{\mathbb{I}}_{\gamma }^2\cap M_{\alpha }}\left\{{\dot{r}}_{\eta }:\eta <\nu \right\}\text{ is a maximal antichain}$$

$$\left\{\mathrm{supp}(p_{\alpha }^L)\cup \mathrm{supp}(p_{\alpha }^R):\alpha \in Y\right\}$$

$$(q\upharpoonright {\gamma }^{\prime },{\dot{1}}_{{\mathbb{I}}_{{\gamma }^{\prime }}^1}){⊩}_{{\mathbb{I}}_{{\gamma }^{\prime }}^2\ast {\dot{\mathbb{I}}}_{{\gamma }^{\prime }}^1}{\dot{x}}^L\perp {\dot{x}}^R$$

$$(p_{{\alpha }_0}^L\upharpoonright {\gamma }^{\prime },{\dot{r}}_{\eta }\upharpoonright {\gamma }^{\prime })$$

$$(p_{{\alpha }_1}^R\upharpoonright {\gamma }^{\prime },{\dot{r}}_{\eta }\upharpoonright {\gamma }^{\prime })$$

$$(q\upharpoonright {\gamma }^{\prime },{\dot{r}}_{\eta }\upharpoonright {\gamma }^{\prime })⊩{\dot{x}}^L{\perp }_{{\dot{T}}_{{\gamma }^{\prime }}}{\dot{x}}^R$$

$$(q\upharpoonright {\gamma }^{\prime },{\dot{1}}_{{\mathbb{I}}_{{\gamma }^{\prime }}^2})⊩{\dot{x}}^L{\perp }_{{\dot{T}}_{{\gamma }^{\prime }}}{\dot{x}}^R$$

$${\square }_{{\omega }_1,2}+\mathsf{SATP}({\aleph }_2)+\mathsf{SATP}({\aleph }_1)$$

$${\vec{\mathbb{I}}}^2$$

$${\dot{\vec{\mathbb{I}}}}^1$$

$$\vec{\mathbb{I}}=⟨⟨{\mathbb{I}}_{\gamma }:\gamma \le \delta ⟩,⟨{\dot{\mathbb{J}}}_{\gamma }:\gamma <\delta ⟩⟩$$

$${\mathbb{I}}_{\gamma }={\mathbb{I}}_{\bar{\gamma }}\ast {\dot{\mathbb{B}}}_{{\mathbb{I}}_{\bar{\gamma }}(<{\kappa }_n)}({\dot{T}}_{\bar{\gamma }})$$

$${\mathbb{I}}_{\gamma }(>{\kappa }_n)=\left\{p\in {\mathbb{I}}_{\gamma }:p(0)\in {\mathbb{I}}_1(>{\kappa }_n)\wedge \mathrm{supp}(p)\setminus \left\{0\right\}\subseteq \bigcup _{m>n}{h}^{-1}(m)\right\}$$

Full text

Acknowledgments

The author would like to thank Itay Neeman for his assistance in revising this paper.

1 Introduction

Given some infinite cardinal $\kappa$, a $\kappa$-Aronszajn tree is a tree of height $\kappa$ without cofinal branches all of whose levels have cardinality $<\kappa$. By classical results of König and Aronszajn, respectively, there are no $\aleph_{0}$-Aronszajn trees but there are $\aleph_{1}$-Aronszajn trees.

Of particular interest to us are special Aronszajn trees. For any successor cardinal $\kappa^{+}$, we say that a $\kappa^{+}$-Aronszajn tree $T$ is special if there exists a function $f\colon T\rightarrow\kappa$ such that if $x<_{T}y$ then $f(x)\neq f(y)$. Following [2], if there are Aronszajn trees of height $\kappa^{+}$ and all such trees are special, we say that the Special Aronszajn Tree Property holds at $\kappa^{+}$, and denote this by $\mathsf{SATP}(\kappa^{+})$.

By a result of Baumgartner, Malitz, and Reinhardt [3] the forcing axiom $\mathsf{MA}_{\aleph_{1}}$ implies $\mathsf{SATP}(\aleph_{1})$. Laver and Shelah [4] showed that $\mathsf{SATP}(\aleph_{2})$ is consistent assuming the existence of a weakly compact cardinal. The forcing which achieves this result is a Levy Collapse of $\kappa$ to $\aleph_{2}$ followed by an iteration of length $\geq\kappa^{+}$ of posets which successively specialize all new $\kappa$-Aronszajn trees arising in the extension.

Golshani and Hayut [2] showed that under the same assumption it is consistent that $\mathsf{SATP}(\aleph_{1})$ and $\mathsf{SATP}(\aleph_{2})$ hold simultaneously, and achieved a global result by showing that it is consistent that $\mathsf{SATP}(\kappa^{+})$ holds simultaneously for all regular $\kappa$, assuming the existence of a proper class of supercompact cardinals. This result is achieved by adapting the methods of [4] to specialize all possible names for trees of height $\kappa^{+}$ while anticipating the specialization of trees of height $\kappa$.

Another class of combinatorial principles of interest to set theorists are square principles. The original square principle $\square_{\kappa}$ was introduced by Jensen [5], who also introduced a weak variant $\square^{*}_{\kappa}$. Later Schimmerling [6] investigated principles $\square_{\kappa,\lambda}$ of intermediate strength. Suppose $\kappa$ is an infinite cardinal and $\lambda$ is a nonzero (but potentially finite) cardinal. A $\square_{\kappa,\lambda}$ sequence is a sequence $\vec{\mathcal{C}}=\langle\mathcal{C}_{\alpha}\colon\alpha\in\operatorname{Lim}{(\kappa^{+})}\rangle$ such that:

1. (1)

For all $\alpha\in\operatorname{Lim}{(\kappa^{+})}$, $1\leq|\mathcal{C}_{\alpha}|\leq\lambda$. 2. (2)

For all $\alpha\in\operatorname{Lim}{(\kappa^{+})}$ and $C\in\mathcal{C}_{\alpha}$, $C$ is club in $\alpha$ and $\operatorname{otp}{C}\leq\kappa$. 3. (3)

For all $\alpha\in\operatorname{Lim}{(\kappa^{+})}$, every $C\in\mathcal{C}_{\alpha}$ threads $\langle\mathcal{C}_{\beta}\colon\beta<\alpha\rangle$ in the sense that $C\cap\beta\in\mathcal{C}_{\beta}$ for all $\beta$ which are limit points of $C$.

We say $\square_{\kappa,\lambda}$ holds if such a sequence exists. Jensen’s original principle $\square_{\kappa}$ is $\square_{\kappa,1}$, and the weak square principle $\square^{*}_{\kappa}$ is $\square_{\kappa,\kappa}$. In this paper we will be concerned exclusively with $\square_{\kappa,2}$.

We show that the result of Laver and Shelah may be improved by establishing the consistency of $\mathsf{SATP}(\aleph_{2})$ with $\square_{\omega_{1},2}$. By a result of Shelah and Stanley [1] $\mathsf{SATP}(\aleph_{2})$ is incompatible with $\square_{\omega_{1}}$, so this result is optimal. Our result is obtained by using an iteration similar to that of Laver-Shelah, with the exception that we use a poset of Cummings and Schimmerling [7]–which collapses weakly compact $\kappa$ to $\aleph_{2}$ while adding a $\square_{\omega_{1},2}$-sequence simultaneously–in place of the Levy Collapse.

Furthermore, we show that our methods are compatible with the anticipatory framework of Golshani and Hayut, and thus we are also able to obtain the analogous global result–namely the consistency of $\mathsf{SATP}(\kappa^{+})$ plus $\square_{\kappa,2}$ for all regular $\kappa$.

2 The Cummings-Schimmerling Poset

We describe here a poset introduced by Cummings and Schimmerling [7] which, given cardinals $\mu<\kappa$ with $\mu$ regular and $\kappa$ inaccessible, will simultaneously collapse $\kappa$ to become $\mu^{+}$ and add a $\square_{\mu,2}$ sequence.

We will denote this poset by $\mathbb{P}(\mu,<\kappa)$. A conditiion $p$ in $\mathbb{P}(\mu,<\kappa)$ is a function such that:

The domain of $p$ is a closed set of ordinals below $\kappa$ of cardinality $<\mu$. 2. 2.

If $\alpha\in\operatorname{dom}{p}$ is a successor ordinal, say $\alpha=\bar{\alpha}+1$, then the unique element of $p(\alpha)$ is $\left\{\bar{\alpha}\right\}$. 3. 3.

If $\alpha\in\operatorname{dom}{p}$ is a limit ordinal with cofinality $<\mu$ then $1\leq|p(\alpha)|\leq 2$ and each element of $p(\alpha)$ is a club subset of $\alpha$ with order type $<\mu$. 4. 4.

If $\alpha\in\operatorname{dom}{p}$ is a limit ordinal with cofinality $\geq\mu$ then $p(\alpha)=\left\{C\right\}$, where $C$ is some closed subset of $\alpha$ with order type $<\mu$ such that $\max{C}=\sup{(\operatorname{dom}{p}\cap\alpha)}$. 5. 5.

(Coherence) If $\alpha\in\operatorname{dom}{p}$, $C\in p(\alpha)$, and $\beta\in C$, then $\beta\in\operatorname{dom}{p}$. If moreover $\beta\in\operatorname{Lim}{C}$, then $C\cap\beta\in p(\beta)$.

The ordering of the poset is defined by $q\leq p$ iff $\operatorname{dom}{q}\supseteq\operatorname{dom}{p}$ and:

1. (a)

$q(\alpha)=p(\alpha)$ for all $\alpha\in\operatorname{dom}{p}$ of cofinality $<\mu$. 2. (b)

If $\alpha$ is of cofinality $\geq\mu$, $p(\alpha)=\left\{C\right\}$, and $q(\alpha)=\left\{D\right\}$, then $C=D\cap(\max{C}+1)$.

The Cummings-Schimmerling poset will not be $<\mu$-closed but will still be sufficiently closed so as to not add bounded subset of $\mu$:

Definition 1**.**

Suppose that $\nu$ is some ordinal. A poset $\mathbb{P}$ is said to be $\nu$-strategically closed if Player II has a winning strategy in the following game $G(\mathbb{P},\nu)$ of length $\nu$:

[TABLE]

In this game the two players alternate building a descending chain $\langle p_{\xi}\colon 1\leq\xi<\nu\rangle$ with Player II playing at all even ordinals (including limits) and Player II loses if he is unable to make a legal move.

Suppose that $\mu$ is some cardinal. We say that $\mathbb{P}$ is $<\mu$-strategically closed if it is $\nu$-strategically closed for all $\nu<\mu$.

Lemma 2**.**

Suppose that $\mu<\kappa$ are cardinals with $\mu$ regular and $\kappa$ inaccessible. Then $\mathbb{P}(\mu,<\kappa)$ is $<\mu$-strategically closed and $\kappa$-Knaster. Moreover, in the generic extension by $\mathbb{P}(\mu,<\kappa)$, $\kappa=\mu^{+}$ and $\square_{\mu,2}$ holds.

Proof.

We prove strategic closure. Fix $\nu<\mu$. We define a winning strategy for Player II in the game $G(\mathbb{P}(\mu,<\kappa),\nu)$. Suppose that $\xi<\nu$ is even and $\langle p_{\zeta}\colon 1\leq\zeta<\xi\rangle$ have already been played.

If $\xi$ is a successor ordinal then Player II plays an arbitrary extension $p_{\xi}$ of $p_{\xi-1}$ such that $\sup{(\operatorname{dom}{(p_{\xi})})}$ is strictly greater than $\sup{(\operatorname{dom}{(p_{\xi-1})})}$. Now suppose that $\xi$ is a nonzero limit ordinal and let

[TABLE]

We define a condition $p_{\xi}$ with domain $\bar{D}=D\cup\operatorname{Lim}{(D)}$ which extends all $\langle p_{\zeta}\colon 1\leq\zeta<\xi\rangle$. First, if $\alpha\in D$ with $\operatorname{cf}{(\alpha)}<\mu$, we let $p_{\xi}(\alpha)=p_{\zeta}(\alpha)$ for any $1\leq\zeta<\xi$ such that $\alpha\in\operatorname{dom}{(p_{\zeta})}$. If $\alpha\in D$ with $\operatorname{cf}{(\alpha)}\geq\mu$ then for each $1\leq\zeta<\xi$ let $C_{\zeta}$ be the unique element of $p_{\alpha}(\zeta)$ and let $p_{\xi}(\alpha)=\left\{C\right\}$, where

[TABLE]

If $\alpha\in\operatorname{Lim}{D}\setminus D$ is below $\sup{D}$ then choose $\beta\in D$ least such that $\alpha<\beta$. Note that $\beta$ must be a limit ordinal by conditions (2), (5) in the definition of $\mathbb{P}(\mu,<\kappa)$. We claim $\operatorname{cf}{(\beta)}\geq\mu$.

Suppose otherwise and let $E$ be an element of $p_{\xi}(\beta)$ (note that $p_{\xi}(\beta)$ has already been defined). If $\alpha$ is not a limit point of $E$, then let $\gamma$ be the least element of $E$ above $\alpha$. Then $\alpha<\gamma<\beta$, and by condition (5) in the definition of $\mathbb{P}(\mu,<\kappa)$ we have $\gamma\in D$, contradicting choice of $\beta$.

Thus $\operatorname{cf}{(\beta)}\geq\mu$ as desired, and so if we let $E$ be the unique element of $p_{\xi}(\beta)$ (again, this has already been defined) then clause (4) in the definition of $\mathbb{P}(\mu,<\kappa)$ guarantees $\max{(E)}=\alpha$, and we may define $p_{\xi}(\alpha)=\left\{E\right\}$.

Finally, if $\alpha=\sup{D}$, let

[TABLE]

It should be clear that $p_{\xi}$ as defined above is a condition in $\mathbb{P}(\mu,<\kappa)$ and the strategy described is a winning strategy for Player II in $G(\mathbb{P}(\mu,<\kappa),\nu)$.

The rest of the lemma may be proved exactly as in [7]. ∎

Note that an argument similar to the one above will show that if $\mu=\aleph_{1}$ then $\mathbb{P}(\mu,<\kappa)$ is in fact countably closed.

In the proof of our consistency results it will be crucial that for $\mu<\kappa_{0}<\kappa_{1}$ with $\mu$ regular and $\kappa_{0}$, $\kappa_{1}$ inacessible, $\mathbb{P}(\mu,<\kappa_{0})$ may be viewed as a factor of $\mathbb{P}(\mu,<\kappa_{1})$. In order to precisely state the necessary factorization result, we first define two auxilliary posets:

Suppose that $\vec{\mathcal{C}}=\langle\mathcal{C}_{\alpha}\colon\alpha<\mu^{+}\rangle$ is a $\square_{\mu,2}$-sequence. We let $\mathbb{T}=\mathbb{T}_{\vec{\mathcal{C}}}$ be the poset of closed bounded $C\subseteq\mu^{+}$ of order-type $<\mu$ such that $C$ threads $\langle\mathcal{C}_{\alpha}\colon\alpha\leq\max{C}\rangle$ in the sense that $C\cap\alpha\in\mathcal{C}_{\alpha}$ for all $\alpha\in\operatorname{Lim}{C}$. For $C,D\in\mathbb{T}$, we set $D\leq C$ if and only if $D$ is an end-extension of $C$.

Finally, if $G$ is the generic added by $\mathbb{P}(\mu,<\kappa_{0})$, then $\mathbb{Q}=\mathbb{Q}_{\mu,\kappa_{0},\kappa_{1},G}$ is the poset defined in $V\left[G\right]$ by setting $q\in\mathbb{Q}$ iff:

1. (1)

The domain of $q$ is a set of limit ordinals in the interval $(\kappa_{0},\kappa_{1})$ of size $<\mu$. 2. (2)

If $\alpha\in\operatorname{dom}{q}$ has cofinality $<\mu$ then $1\leq|q(\alpha)|\leq 2$ and each element of $q(\alpha)$ is a club subset of $\alpha$ with order type $<\mu$. 3. (3)

If $\alpha\in\operatorname{dom}{q}$ has cofinality $\geq\mu$ then $q(\alpha)=\left\{C\right\}$, where $C$ is a club subset of $\alpha$ with order type $<\mu$ such that $\max{C}\geq\sup{(\operatorname{dom}{q}\cap\alpha)}$. 4. (4)

(Coherence) If $\alpha\in\operatorname{dom}{q}$, $C\in q(\alpha)$, and $\beta\in\operatorname{Lim}{C}$, then:

1. (A)

If $\beta>\kappa_{0}$, then $\beta\in\operatorname{dom}{q}$ and $C\cap\beta\in q(\beta)$. 2. (B)

If $\beta<\kappa_{0}$, then $C\cap\beta\in\mathcal{C}_{\beta}$, where $\langle\mathcal{C}_{\beta}\colon\beta<\kappa_{0}\rangle$ is $\bigcup{G}$.

For two elements $p,q\in\mathbb{Q}_{\mu,\kappa_{0},\kappa_{1},G}$, we set $p\leq q$ iff:

1. (1)

$\operatorname{dom}{q}\subseteq\operatorname{dom}{p}$. 2. (2)

For all $\alpha\in\operatorname{dom}{q}$:

1. (a)

If $\alpha$ has cofinality $<\mu$ then $p(\alpha)=q(\alpha)$. 2. (b)

If $\alpha$ has cofinality $\geq\mu$, $p(\alpha)=\left\{C\right\}$, and $q(\alpha)=\left\{D\right\}$, then $C$ is an end-extension of $D$.

Lemma 3**.**

Suppose that $\mu<\kappa_{0}<\kappa_{1}$ are cardinals with $\mu$ regular and $\kappa_{0}$, $\kappa_{1}$ inaccessible, and $\dot{G}$ is the canonical name for the $\mathbb{P}(\mu,<\kappa_{0})$-generic. Then if we let $\dot{\mathbb{T}}=\check{\mathbb{T}}_{\bigcup{\dot{G}}}$, $\dot{\mathbb{Q}}=\check{\mathbb{Q}}_{\mu,\kappa_{0},\kappa_{1},\dot{G}}$, there is an isomorphism between a dense subset of $\mathbb{P}(\mu,<\kappa_{1})$ and a dense subset of $\mathbb{P}(\mu,<\kappa_{0})\ast\dot{\mathbb{T}}\ast\dot{\mathbb{Q}}$. In particular these two forcings are equivalent, and so informally we may view them as being equal.

Proof.

As in [7]. ∎

Lemma 4**.**

Suppose that $\mu<\kappa$ are cardinals with $\mu$ regular and $\kappa$ inaccessible. Let $\mathbb{P}=\mathbb{P}(\mu,<\kappa)$, let $\dot{G}$ be the canonical name for the $\mathbb{P}$-generic, and let $\dot{\mathbb{T}}=\check{\mathbb{T}}_{\bigcup\dot{G}}$. Then there is a dense subset of $\mathbb{P}\ast\dot{\mathbb{T}}$ which is $<\mu$-closed and so in particular $\mathbb{P}\ast\dot{\mathbb{T}}$ is forcing equivalent to $\operatorname{Col}{(\mu,\kappa)}$.

Proof.

Let $D$ be the dense set of conditions in $\mathbb{P}\ast\dot{\mathbb{T}}$ of the form $(p,\check{t})$ which are flat in the sense that $\max{(\operatorname{dom}{p})}=\max{t}$. We claim that $D$ is as desired. To see this, suppose that $\nu<\mu$ is a limit ordinal and let $\langle(p_{\xi},\check{t}_{\xi})\colon\xi<\nu\rangle$ be a descending sequence of conditions in $D$. We find a lower bound $p^{*}$ for $\langle p_{\xi}\colon\xi<\nu\rangle$ as in the limit case of Lemma 2, except that we set

[TABLE]

where

[TABLE]

Then $(p^{*},\check{t}^{*})\in D$ is our desired lower bound. Since $D$ is $<\mu$-closed, $|D|=\kappa$, and $D$ forces $|\kappa|=\mu$, $D$ is forcing equivalent to $\operatorname{Col}{(\mu,\kappa)}$ by a well-known result due to Solovay (see, e.g. Lemma 2.3 of [8]). ∎

3 Specializing Trees with Anticipation

In this section we review the methods of [2] for specializing trees while anticipating subsequent forcing.

First we introduce the modified Baumgartner forcing which specializes a single tree while anticipating a single subsequent forcing.

Definition 5** ([2]).**

Suppose that $\mu<\kappa$ are regular cardinals in $V$ and $\mathbb{I}^{2}\ast\dot{\mathbb{I}}^{1}$ is a $\kappa$-c.c. two-step iteration which forces $\kappa=\mu^{+}$. Suppose moreover that $\dot{T}$ is an $\mathbb{I}^{2}\ast\dot{\mathbb{I}}^{1}$-name for a $\kappa$-Aronszajn tree, which we view as a subset of $\kappa\times\mu$. Then $\mathbb{B}_{\mu,\mathbb{I}^{1}}\!(\dot{T})$ is defined in $V^{\mathbb{I}^{2}}$ as the poset of partial functions $f\colon\kappa\times\mu\rightarrow\mu$ of size $<\mu$ such that if $s,t\in\operatorname{dom}{f}$ and $f\!\left(s\right)=f\!\left(t\right)$, then

[TABLE]

The forcing is ordered by reverse inclusion.

If $\mu$ is understood (as it usually is) then we suppress the dependence on $\mu$ and write $\mathbb{B}_{\mathbb{I}^{1}}(\dot{T})$ in place of $\mathbb{B}_{\mu,\mathbb{I}^{1}}(\dot{T})$.

Lemma 6** ([2]).**

Suppose $\mu<\kappa$ are regular cardinals and, $\mathbb{I}^{2}\ast\dot{\mathbb{I}}^{1}$ is a $\kappa$-c.c. forcing which forces $\kappa=\mu^{+}$, $\dot{T}$ is an $\mathbb{I}^{2}\ast\mathbb{I}^{1}$-name for a $\kappa$-Aronszajn tree, and $G$ is $\mathbb{I}^{2}$-generic. Then in $V\!\left[G\right]$ the following hold:

1. (a)

$\mathbb{B}_{\mathbb{I}^{1}}(\dot{T})$* is $<\mu$-closed.* 2. (b)

In the extension by the generic for $\mathbb{B}_{\mathbb{I}^{1}}(\dot{T})$ there is a function $F\colon\kappa\times\mu\rightarrow\mu$ which is a specializing function for the tree $\dot{T}\left[G\right]\left[H\right]$ for any $\mathbb{I}^{1}$-generic $H$.

Now we describe the general form of iterations $\vec{\mathbb{I}}^{2}$ and $\vec{\mathbb{I}}^{1}$ such that $\vec{\mathbb{I}}^{2}$ specializes all $\kappa$-Aronszajn trees while anticipating forcing by $\vec{\mathbb{I}}^{1}$.

Definition 7**.**

Suppose that $\mu<\kappa<\kappa^{+}\leq\delta$ are regular cardinals in $V$, and the iterations

[TABLE]

are as follows:

• •

$\mathbb{I}^{2}_{1}=\mathbb{P}(\mu,<\kappa)$, the forcing which collapses $\kappa$ to $\mu^{+}$ while adding $\square_{\mu,2}$.

• •

$\mathbb{I}^{2}_{\gamma}$ is the iteration with $<\mu$-support of $\langle\dot{\mathbb{J}}^{2}_{\gamma^{\prime}}\colon\gamma^{\prime}<\gamma\rangle$. In other words, if $\gamma$ is a limit ordinal of cofinality $\geq\mu$, then $\mathbb{I}^{2}_{\gamma}$ is the direct limit of $\langle\mathbb{I}^{2}_{\gamma^{\prime}}\colon\gamma^{\prime}<\gamma\rangle$, if $\gamma$ is a limit ordinal of cofinality $<\mu$, then $\mathbb{I}^{2}_{\gamma}$ is the inverse limit of $\langle\mathbb{I}^{2}_{\gamma^{\prime}}\colon\gamma^{\prime}<\gamma\rangle$, and if $\gamma=\bar{\gamma}+1$ is a successor ordinal then $\mathbb{I}^{2}_{\gamma}=\mathbb{I}^{2}_{\bar{\gamma}}\ast\dot{\mathbb{J}}^{2}_{\bar{\gamma}}$.

• •

Each $\dot{\mathbb{I}}^{1}_{\gamma}$ is an $\mathbb{I}^{2}_{\gamma}$-name for a $\mu$-c.c. poset.

• •

$\dot{\mathbb{J}}^{2}_{\gamma}$ is a name for the poset $\mathbb{B}_{\mathbb{I}^{1}_{\gamma}}(\dot{T}_{\gamma})$, where $\dot{T}_{\gamma}$ is an $\mathbb{I}^{2}_{\gamma}\ast\dot{\mathbb{I}}^{1}_{\gamma}$-name for a $\kappa$-Aronszajn tree, chosen according to some appropriate bookkeeping function.

Then we refer to $\vec{\mathbb{I}}^{2}$ as an “iteration which collapses $\kappa$ to $\mu^{+}$, adds $\square_{\mu,2}$ and specializes all $\kappa$-Aronszajn trees while anticipating the subsequent iteration $\vec{\mathbb{I}}^{1}$” (or some similar locution for the sake of brevity).

Definition 8** ([2]).**

Suppose that $\mu<\kappa<\delta$ are regular cardinals and $\vec{\mathbb{I}}^{2}$ is an iteration of length $\delta$ which collapses $\kappa$ to $\mu^{+}$, adds $\square_{\mu,2}$, and anticipates the iteration $\dot{\vec{\mathbb{I}}}^{1}$ in the sense described above. Let $\gamma\leq\delta$ be some ordinal and suppose that $M$ is an elementary substructure of $H(\theta)$ ($\theta$ sufficiently large) of cardinality $\kappa$ such that $V_{\kappa}\cup M^{<\kappa}\cup\left\{\gamma\right\}\subseteq M$ and $M$ contains all relevant parameters. Furthermore, let $\phi\colon\kappa\rightarrow M$ be a bijection and for all $\alpha<\kappa$ set $M_{\alpha}=\phi``\alpha$. We say that $\vec{\mathbb{I}}^{2}$, $\dot{\vec{\mathbb{I}}}^{1}$ are suitable for $M$, $\phi$, $\gamma$ if:

1. (1)

For all $\bar{\gamma}\leq\gamma$, $\Vdash_{\mathbb{I}^{2}_{\gamma}}$ “$\dot{\mathbb{I}}^{1}_{\bar{\gamma}}$ is $\mu$-c.c.” 2. (2)

For all $\alpha<\kappa$ and $\bar{\gamma}\in M_{\alpha}\cap\gamma$, if:

1. (a)

$\mathbb{I}^{2}_{\bar{\gamma}}\cap M_{\alpha}$ is a regular subposet of $\mathbb{I}^{2}_{\bar{\gamma}}\cap M$. 2. (b)

$\Vdash_{\mathbb{I}^{2}_{\bar{\gamma}}\cap M}$ “$\dot{\mathbb{I}}^{1}_{\bar{\gamma}}\cap M_{\alpha}$ is a regular subiteration of $\dot{\mathbb{I}}^{1}_{\bar{\gamma}}\cap M$.” 3. (c)

$\dot{T}_{\bar{\gamma}}\cap M_{\alpha}$ is an $(\mathbb{I}^{2}_{\bar{\gamma}}\cap M_{\alpha})\ast(\dot{\mathbb{I}}^{1}_{\bar{\gamma}}\cap M_{\alpha})$-name for an $\alpha$-Aronszajn tree.

Then forcing with $(\mathbb{I}^{2}_{\bar{\gamma}}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\bar{\gamma}}\cap M)/G$, where $G$ is generic for $(\mathbb{I}^{2}_{\bar{\gamma}}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\bar{\gamma}}\cap M_{\alpha})$, doesn’t add any new branches to the tree named by $\dot{T}_{\bar{\gamma}}\cap M_{\alpha}$.

We will need to make use of the following basic lemma about forcings which don’t add branches to trees:

Lemma 9** (Folklore, see [9], [10]).**

Suppose that $T$ is a $\kappa$-tree and $\mathbb{P}$ is a $\kappa$-Knaster poset. Then forcing with $\mathbb{P}$ doesn’t add a branch to $T$.

4 Obtaining $\square_{\omega_{1},2}$ + $\mathsf{SATP}(\aleph_{2})$ +

$\mathsf{SATP}(\aleph_{1})$

Theorem 10**.**

Suppose that $\mu<\kappa<\kappa^{+}\leq\delta$ are cardinals with $\mu$, $\delta$ regular and $\kappa$ weakly compact. Suppose moreover that

[TABLE]

are two iterations such that $\vec{\mathbb{I}}^{2}$ collapses $\kappa$ to $\mu^{+}$, adds $\square_{\mu,2}$, and specializes all $\kappa$-Aronszajn trees while anticipating $\vec{\mathbb{I}}^{1}$ (in the sense described in the previous section).

Finally, suppose that for all ordinals $\gamma\leq\delta$ there exists $M$ elementary in $H(\theta)$ ($\theta$ sufficiently large) of cardinality $\kappa$ such that $V_{\kappa}\cup M^{<\kappa}\cup\left\{\gamma\right\}\subseteq M$, $M$ contains all relevant parameters, and $\vec{\mathbb{I}}^{2}$, $\dot{\vec{\mathbb{I}}}^{1}$ are suitable for $M$, $\phi$, $\gamma$ (for some fixed bijection $\phi\colon\kappa\rightarrow M$). Then the generic extension by $\mathbb{I}^{2}_{\delta}\ast\dot{\mathbb{I}}^{1}_{\delta}$ satisfies

[TABLE]

The majority of the remainder of this section is devoted to giving a proof of this result. We follow closely the proof of the main theorem in [2].

Lemma 11**.**

For every $\gamma\leq\delta$, $\mathbb{I}^{2}_{\gamma}$ is $<\mu$ strategically closed.

Proof.

The forcing $\mathbb{I}^{2}_{\gamma}$ is a $<\mu$-strategically closed forcing (namely, $\mathbb{P}(\mu,<\kappa)$) followed by the $<\mu$-support iteration of $<\mu$-closed posets. ∎

Lemma 12**.**

For every $\gamma\leq\delta$, $\Vdash_{\mathbb{I}^{2}_{\gamma}}``\dot{\mathbb{I}}^{1}_{\gamma}$ is $\mu$-c.c.” and $\Vdash_{\mathbb{I}^{2}_{\delta}}``\dot{\mathbb{I}}^{1}_{\gamma}$ is $\mu$-c.c.”

Proof.

This is immediate from the definition of suitability of $\vec{\mathbb{I}}^{2}$, $\dot{\vec{\mathbb{I}}}^{1}$. ∎

Lemma 13**.**

For every $\gamma\leq\delta$, $\mathbb{I}^{2}_{\gamma}$ is $\kappa$-Knaster.

Proof.

By induction on $\gamma$. For the base case, we know $\mathbb{I}^{2}_{1}\simeq\mathbb{P}\!\left(\mu,<\kappa\right)$ is $\kappa$-Knaster by Lemma 2. So suppose $\gamma\leq\delta$ and each $\mathbb{I}^{2}_{\gamma^{\prime}}$ is $\kappa$-Knaster for all $\gamma^{\prime}<\gamma$. We seek to show that $\mathbb{I}^{2}_{\gamma}$ is also $\kappa$-Knaster.

If $\gamma$ is a limit ordinal and $\mu\leq\operatorname{cf}{\gamma}\neq\kappa$ this is immediate since any subset of $\mathbb{I}^{2}_{\gamma}$ of cardinality $\kappa$ may be refined to a subset of $\mathbb{I}^{2}_{\gamma^{\prime}}$ of cardinality $\kappa$ for some $\gamma^{\prime}<\gamma$.

If $\gamma$ is a limit ordinal with $\operatorname{cf}{\gamma}=\kappa$ this follows from a $\Delta$-system argument.

Thus suppose that either $\gamma$ is a limit ordinal with $\operatorname{cf}{\gamma}<\mu$ or $\gamma=\bar{\gamma}+1$ for some ordinal $\bar{\gamma}$. Fix $M$ as in the hypothesis of the theorem, and in either case fix an increasing sequence $\left\{\gamma_{i}\colon i<\operatorname{cf}{\gamma}\right\}$ in $M$ which is cofinal in $\gamma$ (if $\gamma=\bar{\gamma}+1$ is a successor ordinal we say its cofinality is $1$ and we let $\gamma_{0}=\bar{\gamma}$, so in this case $\left\{\gamma_{i}\colon i<1\right\}$ is cofinal in $\gamma$).

Let $R$ be a subset of $V_{\kappa}$ which encodes both $M$ and $\phi$ (where $\phi$ is the bijection from the hypothesis of Theorem 10). Fix a $<\kappa$-complete normal filter $\mathcal{F}$ on $\kappa$ which extends the club filter and satisfies

[TABLE]

for each formula $\psi$ which is $\Pi^{1}_{1}$ over $V_{\kappa}$. For all $\alpha<\kappa$ set $M_{\alpha}=\phi``\alpha$.

Claim 14** ([4], [2]).**

Assume that $\bar{\gamma}\in\gamma\cap M$, and for all $\bar{\bar{\gamma}}\in\bar{\gamma}\cap M$ we have that $\mathbb{I}^{2}_{\bar{\bar{\gamma}}}$ is $\kappa$-Knaster and $\dot{T}_{\bar{\bar{\gamma}}}$ is an $\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}}$-name for a $\kappa$-Aronszajn tree. Then there exists $X=X_{\bar{\gamma}}\in\mathcal{F}$ such that for all $\alpha\in X$ and $\bar{\bar{\gamma}}\in\bar{\gamma}\cap M_{\alpha}$:

$\alpha$* is inaccessible.* 2. 2.

$M_{\alpha}\cap\kappa=\alpha$. 3. 3.

$M_{\alpha}^{<\alpha}\subseteq M_{\alpha}$. 4. 4.

$\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\cap M_{\alpha}$* is a regular subposet of $\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\cap M$ and is $\alpha$-c.c.* 5. 5.

$\mathbb{I}^{1}_{\bar{\bar{\gamma}}}\cap M_{\alpha}$* is equivalent to an $\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\cap M_{\alpha}$-name.* 6. 6.

$(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M_{\alpha}$* is a regular subposet of $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M$.* 7. 7.

$(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M_{\alpha}$* forces that $T_{\bar{\bar{\gamma}}}\cap(\alpha\times\mu)$ is an $\alpha$-Aronszajn tree.*

Proof.

Let $X=X_{\bar{\gamma}}$ be the set of all $\alpha<\kappa$ that satisfy these requirements for all $\bar{\bar{\gamma}}\in\bar{\gamma}\cap M_{\alpha}$. The claim follows immediately from a $\Pi^{1}_{1}$ reflection argument together with the fact that $\mathcal{F}$ extends the club filter. In (7) we make use of the fact that $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M$ forces that $T_{\bar{\bar{\gamma}}}$ is a $\kappa$-Aronszajn tree, which follows from observing that $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M$ is a regular subposet of $\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}}$ (this is itself a consequence of the fact that $\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}}$ has the $\kappa$-c.c., by the inductive hypothesis). ∎

Definition 15** ([4]).**

A condition $p\in\mathbb{I}^{2}_{\gamma}$ is said to be determined if there is in $V$ a sequence $\langle x_{\xi}\colon 1\leq\xi<\gamma\rangle$ such that for all $1\leq\xi<\gamma$, $p\restriction\xi\Vdash_{\mathbb{I}^{2}_{\xi}}p\left(\xi\right)=\check{x}_{\xi}$.

As in [4], we may easily observe that the set of determined conditions is dense in $\mathbb{I}^{2}_{\gamma}$.

Definition 16** ([2]).**

Suppose that $p\in\mathbb{I}^{2}_{\gamma}\cap M$ is some condition and $\alpha<\kappa$. Write $p=\langle p(\xi)\colon\xi<\gamma\rangle$. Then $p\restriction M_{\alpha}$ denotes the condition $\langle p^{\prime}(\xi)\colon\xi<\gamma\rangle$, where $p^{\prime}(\xi)$ is the trivial condition if $\xi\notin M_{\alpha}$ and $p^{\prime}(\xi)=p(\xi)\cap M_{\alpha}$ otherwise. We say that $p$ is $\alpha$-compatible if $p\restriction M_{\alpha}$ forces that $p$ is a determined condition in $(\mathbb{I}^{2}_{\gamma}\cap M)/(G_{\mathbb{I}^{2}_{\gamma}}\cap M_{\alpha})$.

Claim 17**.**

Let $X=X_{\bar{\gamma}}$ be as in the previous claim. Then for every $\alpha\in X$, $\bar{\bar{\gamma}}\in(\bar{\gamma}+1)\cap M_{\alpha}$, $\bar{p}\in\mathbb{I}^{2}_{\gamma}\cap M_{\alpha}$, $\alpha$-compatible $p^{L},p^{R}\in\mathbb{I}^{2}_{\gamma}\cap M$ with $\bar{p}=p^{L}\restriction M_{\alpha}=p^{R}\restriction M_{\alpha}$, and every pair $(\dot{x}^{L},\dot{x}^{R})$ of $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\cap M)\ast(\mathbb{I}^{1}_{\bar{\bar{\gamma}}}\cap M_{\alpha})$-names for nodes in $T_{\bar{\bar{\gamma}}}$ above level $\alpha$, there are $\alpha$-compatible conditions $p^{L}_{*},p^{R}_{*}\in\mathbb{I}^{2}_{\gamma}\cap M$, $\bar{p}_{*}\in\mathbb{I}^{2}_{\gamma}\cap M_{\alpha}$, and a sequence

[TABLE]

(for some $\vartheta<\mu$) in $M_{\alpha}$ such that:

1. (a)

$p^{L}_{*}\leq p^{L}$, $p^{R}_{*}\leq p^{R}$ and $\bar{p}_{*}=p^{L}_{*}\restriction M_{\alpha}=p^{R}_{*}\restriction M_{\alpha}$. 2. (b)

For all $\eta<\vartheta$ $\bar{p}_{*}\Vdash_{\mathbb{I}^{2}_{\gamma}\cap M_{\alpha}}r_{\eta}\in\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha}$. 3. (c)

For all $\eta<\vartheta$, $\xi_{\eta}<\alpha$ and $x^{L}_{\eta}$, $x^{R}_{\eta}$ are elements of $\left\{\xi_{\eta}\right\}\times\mu$ with $x^{L}_{\eta}\neq x^{R}_{\eta}$. 4. (d)

$(p^{L}_{*}\restriction\bar{\bar{\gamma}},r_{\eta}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{L}_{\eta}\leq\dot{x}^{L}$* and $(p^{R}_{*}\restriction\bar{\bar{\gamma}},r_{\eta}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{R}_{\eta}\leq\dot{x}^{R}$.* 5. (e)

$\bar{p}_{*}\Vdash_{\mathbb{I}^{2}_{\gamma}\cap M_{\alpha}}\left\{\dot{r}_{\eta}\colon\eta<\vartheta\right\}$* is a maximal antichain in $\dot{\mathbb{I}}^{1}_{\gamma}$.*

Proof.

Suppose $\alpha\in X$, $\bar{\bar{\gamma}}\in(\bar{\gamma}+1)\cap M_{\alpha}$, and fix names $\dot{x}^{L}$, $\dot{x}^{R}$ for nodes in $T_{\bar{\bar{\gamma}}}$ of level $\geq\alpha$ and conditions $\bar{p}$, $p^{L}$, $p^{R}$ as in the statement of the claim. It follows from the choice of $\alpha$ that for any $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}}\cap M_{\alpha})$-generic $G$ the branches in $T_{\bar{\bar{\gamma}}}\restriction\alpha$ below $\dot{x}^{L}$, $\dot{x}^{R}$ are not in $V[G\cap(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\mathbb{I}^{1}_{\bar{\bar{\gamma}}})\cap M_{\alpha}]$.

Subclaim 18**.**

For any pair $(s^{L},t)$, $(s^{R},t)$ of conditions in $(\mathbb{I}^{2}_{\gamma}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha})$ such that $s^{L}$, $s^{R}$ are $\alpha$-compatible and $s^{L}\restriction M_{\alpha}=s^{R}\restriction M_{\alpha}$, there is another pair $(q^{L},r)$, $(q^{R},r)$ of conditions in $(\mathbb{I}^{2}_{\gamma}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha})$ such that:

• •

$(q^{L},r)\leq(s^{L},t)$ and $(q^{R},r)\leq(s^{R},t)$

• •

$(q^{L}\restriction\bar{\bar{\gamma}},r\restriction\bar{\bar{\gamma}})$, $(q^{R}\restriction\bar{\bar{\gamma}},r\restriction\bar{\bar{\gamma}})$ force incompatible values for the branches below $\dot{x}^{L}$ and $\dot{x}^{R}$

• •

$q^{L}$, $q^{R}$ are $\alpha$-compatible

• •

$q^{L}\restriction M_{\alpha}=q^{R}\restriction M_{\alpha}$.

Proof.

This is done exactly as in [2]. We give the proof for the convenience of the reader. Suppose the opposite for the sake of a contradiction, and consider pairs $(s^{L},t)$, $(s^{R},t)$ witnessing the negation. Let $H$ be $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M_{\alpha}$-generic with $(s^{L}\restriction\bar{\bar{\gamma}})\restriction M_{\alpha}=(s^{R}\restriction\bar{\bar{\gamma}})\restriction M_{\alpha}\in H$ and $J_{i}$ be $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\cap M)/(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\cap H)$-mutually generic with $(s^{i}\restriction\bar{\bar{\gamma}},t)\in J_{i}$ (for $i\in\left\{L,R\right\}$).

If $K_{i}$ is any $[(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M]/(H\ast J_{i})$- generic ($i\in\left\{L,R\right\}$) then in $V[H][J_{i}][K_{i}]$ there is a branch $b^{i}$ in the tree $T_{\bar{\bar{\gamma}}}\cap\left(\alpha\times\mu\right)$ consisting of nodes which lie below $x^{i}$. Moreover, by condition (2) of Definition 8, we have $b^{i}\in V[H][J_{i}]$ (note, however, that $V[H][J_{i}]$ may not recognize that all nodes in $b^{i}$ are below $x^{i}$, or even that $T_{\bar{\bar{\gamma}}}$ itself is a tree). Nonetheless, by condition (1) of Definition 8 there exists $\mu_{0}^{i}<\mu$ and a collection $\{\check{b}^{i}_{\xi}\colon\xi<\mu_{0}^{i}\}$ of names for elements of $V[H][J_{i}]$ which are cofinal branches through $T_{\bar{\bar{\gamma}}}\cap(\alpha\times\mu)$ such that in $V[H][J_{i}]$ the following holds:

[TABLE]

where $\dot{b}^{i}$ is the canonical name for the branch $b^{i}$ described above.

Moreover, by the assumption of the subclaim, there must exist $\xi^{L}<\mu_{0}^{L}$ and $\xi^{R}<\mu_{0}^{R}$ such that $b^{L}_{\xi^{L}}=b^{R}_{\xi^{R}}$. Denoting this common value by $b$, we have

[TABLE]

and since $J_{L}$, $J_{R}$ were chosen to be mutually generic we have $b\in V[H]$. But this is a contradiction since $(\mathbb{I}^{2}_{\bar{\bar{\gamma}}}\ast\dot{\mathbb{I}}^{1}_{\bar{\bar{\gamma}}})\cap M_{\alpha}$ forces $T_{\bar{\bar{\gamma}}}\cap\left(\alpha\times\mu\right)$ to be an $\alpha$-Aronszajn tree. ∎

Invoking this claim, we may find a pair of conditions $(p^{L}_{0},r_{0})$, $(p_{0}^{R},r_{0})$ in $(\mathbb{I}^{2}_{\gamma}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha})$ with $p_{0}^{L}\leq p^{L}$, $p^{R}_{0}\leq p^{R}$, and $p_{0}^{L}\restriction M_{\alpha}=p_{0}^{R}\restriction M_{\alpha}$ together with $\xi_{0}<\alpha$ and elements $x_{0}^{L}$, $x_{0}^{R}$ in $\left\{\xi_{0}\right\}\times\mu$ such that

[TABLE]

Furthermore, we may assume that if we let $t_{0}^{L}$ be the unique element of $p_{0}^{L}\!\left(0\right)\!\left(\alpha\right)$ and $t_{0}^{R}$ be the unique element of $p_{0}^{R}\!\left(0\right)\!\left(\alpha\right)$ then

[TABLE]

are flat conditions in $\mathbb{P}\!\left(\mu,<\alpha\right)\ast\dot{\mathbb{T}}_{\alpha}$, where $\dot{\mathbb{T}}_{\alpha}=\check{\mathbb{T}}_{\bigcup{\dot{G}}_{\mathbb{P}(\mu,<\alpha)}}$.

Proceeding inductively, suppose $\nu<\mu$ and we have defined the pairs $(p^{L}_{\eta},\dot{r}_{\eta})$, $(p^{R}_{\eta},\dot{r}_{\eta})$ in $(\mathbb{I}^{2}_{\gamma}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha})$, $\bar{p}_{\eta}$ in $\mathbb{I}^{2}_{\gamma}\cap M_{\alpha}$, and $\check{t}^{L}_{\eta}$, $\check{t}^{R}_{\eta}$ in $\dot{\mathbb{T}}_{\alpha}\cap M$ together with $\xi_{\eta}$ and $x^{L}_{\eta}$, $x^{R}_{\eta}\in\left\{\xi_{\eta}\right\}\times\mu$, such that:

• •

The sequences $\langle p^{L}_{\eta}\colon\eta<\nu\rangle$ and $\langle p^{R}_{\eta}\colon\eta<\nu\rangle$ are decreasing and for each $\eta$ $p^{L}_{\eta}$ and $p^{R}_{\eta}$ are $\alpha$-compatible.

• •

$\bar{p}_{\eta}=p^{L}_{\eta}\restriction M_{\alpha}=p^{R}_{\eta}\restriction M_{\alpha}$.

• •

$\bar{p}_{\eta}\Vdash_{\mathbb{I}^{2}_{\gamma}\cap M_{\alpha}}\dot{r}_{\eta}\in\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha}$.

• •

For $\eta_{0}<\eta_{1}<\nu$, $\bar{p}_{\eta_{1}}\Vdash_{\mathbb{I}^{2}_{\delta}\cap M_{\alpha}}$ $\dot{r}_{\eta_{0}}$, $\dot{r}_{\eta_{1}}$ are incompatible.

• •

$\xi_{n}<\alpha$, $x^{L}_{\eta}$, $x^{R}_{\eta}\in\left\{\xi_{\eta}\right\}\times\mu$ and $x^{L}_{\eta}\neq x^{R}_{\eta}$.

• •

$(p^{L}_{\eta}\restriction\bar{\bar{\gamma}},\dot{r}_{\eta}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{L}_{\eta}\leq\dot{x}^{L}$.

• •

$(p^{R}_{\eta}\restriction\bar{\bar{\gamma}},\dot{r}_{\eta}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{R}_{\eta}\leq\dot{x}^{R}$.

• •

$t^{L}_{\eta}$ is the unique element of $p^{L}_{\eta}\!\left(0\right)\!\left(\alpha\right)$.

• •

$t^{R}_{\eta}$ is the unique element of $p^{R}_{\eta}\!\left(0\right)\!\left(\alpha\right)$.

• •

$(p^{L}_{\eta}\restriction M_{\alpha})\!\left(0\right)\ast\check{t}^{L}_{\eta}$, $(p^{R}_{\eta}\restriction M_{\alpha})\!\left(0\right)\ast\check{t}^{R}_{\eta}$ are flat conditions in $\mathbb{P}(\mu,<\alpha)\ast\dot{\mathbb{T}}_{\alpha}$, where $\dot{\mathbb{T}}_{\alpha}=\check{\mathbb{T}}_{\bigcup\dot{G}_{\mathbb{P}(\mu,<\alpha)}}$.

If $\nu$ is a successor ordinal let $q_{\nu}^{L}=p^{L}_{\nu-1}$, $q^{R}_{\nu}=p^{R}_{\nu-1}$. Otherwise, let

[TABLE]

and let $q^{L}_{\nu}$, $q^{R}_{\nu}$ be lower bounds of $\left\{p^{L}_{\eta}\colon\eta<\nu\right\}$, $\left\{p^{R}_{\eta}\colon\eta<\nu\right\}$ such that both $t^{L}_{\nu}$, $t^{R}_{\nu}$ appear on the (approximations to) square sequences $q^{L}_{\nu}\!\left(0\right)$, $q^{R}_{\nu}\!\left(0\right)$. These lower bounds may be seen to exist by an argument similar to that used to prove strategic closure in Lemma 2. Namely, each initial segment of $t^{L}_{\eta}$, $t^{R}_{\eta}$ of limit order type has already been placed on $p^{L}_{\eta}(0)$, $p^{R}_{\eta}(0)$ for some $\eta<\nu$, and therefore we may place $t^{L}_{\nu}$, $t^{R}_{\nu}$ on $q^{L}_{\nu}(0)$, $q^{R}_{\nu}(0)$ without any danger of violating coherence. Observe that this is the part of the argument where we exploit the “two-ness” of the principle $\square_{\kappa,2}$ (and hence of the poset used to force it). Namely, we seek to ensure that $p^{L}_{*}$, $p^{R}_{*}$ agree on $M_{\alpha}$, and so must put both threads on both conditions. Finally, note that in either case ($\nu$ successor or limit) we have $q_{\nu}^{L},q_{\nu}^{R}$ are $\alpha$-compatible conditions in $\mathbb{I}^{2}_{\gamma}\cap M$ and $q^{L}_{\nu}\restriction M_{\alpha}=q^{R}_{\nu}\restriction M_{\alpha}$. Let $\bar{q}_{\nu}=q^{L}_{\nu}\restriction M_{\alpha}=q^{R}_{\nu}\restriction M_{\alpha}$. If

[TABLE]

we halt the construction and set $p^{L}_{\nu}=q_{\nu}^{L}$, $p^{R}_{\nu}=q^{R}_{\nu}$. Otherwise proceed exactly as when obtaining $r_{0}$, except now working below $s_{\nu}$. Namely, find a condition $s_{\nu}$ forced to be incompatible with every $r_{\eta}$ ($\eta<\nu$) and choose $(p^{L}_{\nu},r_{\nu})$, $(p^{R}_{\nu},r_{\nu})$, $\bar{p}_{\nu}$, $\xi_{\nu}<\alpha$, and $x^{L}_{\nu},x^{R}_{\nu}\in\left\{\xi_{\nu}\right\}\times\mu$ such that:

• •

$(p^{L}_{\nu},r_{\nu}),(p^{R}_{\nu},r_{\nu})\in(\mathbb{I}^{2}_{\gamma}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha})$.

• •

$p^{L}_{\nu}$, $p^{R}_{\nu}$ are $\alpha$-compatible.

• •

$(p^{L}_{\nu},r_{\nu})\leq(q^{L}_{\nu},s_{\nu})$ and $(p^{R}_{\nu},r_{\nu})\leq(q^{R}_{\nu},s_{\nu})$.

• •

$\bar{p}_{\nu}=p^{L}_{\nu}\restriction M_{\alpha}=p^{R}_{\nu}\restriction M_{\alpha}$.

• •

$(p^{L}_{\nu}\restriction\bar{\bar{\gamma}},\dot{r}_{\nu}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{L}_{\nu}\leq\dot{x}^{L}$.

• •

$(p^{R}_{\nu}\restriction\bar{\bar{\gamma}},\dot{r}_{\nu}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{R}_{\nu}\leq\dot{x}^{R}$.

• •

$t_{\nu}^{L}$ is the unique element of $p_{\nu}^{L}(0)(\alpha)$.

• •

$t_{\nu}^{R}$ is the unique element of $p_{\nu}^{R}(0)(\alpha)$.

• •

$(p_{\nu}^{L}\restriction M_{\alpha})(0)\ast\check{t}^{L}_{\nu}$, $(p_{\nu}^{R}\restriction M_{\alpha})(0)\ast\check{t}^{R}_{\nu}$ are flat conditions in $\mathbb{P}(\mu,<\alpha)\ast\dot{\mathbb{T}}_{\alpha}$, where $\dot{\mathbb{T}}_{\alpha}=\check{\mathbb{T}}_{\bigcup\dot{G}_{\mathbb{P}(\mu,<\alpha)}}$

By Lemma 12 this process terminates after $<\mu$ many steps. At its completion we get an ordinal $\vartheta<\mu$, descending sequences $\langle p^{L}_{\eta}\colon\eta\leq\vartheta\rangle$ and $\langle p^{R}_{\eta}\colon\eta\leq\vartheta\rangle$ of conditions in $\mathbb{I}^{2}_{\gamma}\cap M$, as well as sequences $\langle\bar{p}_{\eta}\colon\eta<\vartheta\rangle$, $\langle r_{\eta}\colon\eta<\vartheta\rangle$, and $\langle(\xi_{\eta},\check{t}^{L}_{\eta},\check{t}^{R}_{\eta},\check{x}^{L}_{\eta},\check{x}^{R}_{\eta})\colon\eta<\vartheta\rangle$ such that:

• •

$(p^{L}_{\eta},r_{\eta}),(p^{R}_{\eta},r_{\eta})\in(\mathbb{I}^{2}_{\gamma}\cap M)\ast(\dot{\mathbb{I}}^{1}_{\gamma}\cap M_{\alpha})$.

• •

$p^{L}_{\eta}$, $p^{R}_{\eta}$ are $\alpha$-compatible.

• •

$\bar{p}_{\eta}=p^{L}_{\eta}\restriction M_{\alpha}=p^{R}_{\nu}\restriction M_{\alpha}$.

• •

$x^{L}_{\eta},x^{R}_{\eta}\in\left\{\xi_{\eta}\right\}\times\mu$.

• •

$(p^{L}_{\eta}\restriction\bar{\bar{\gamma}},\dot{r}_{\eta}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{L}_{\eta}\leq\dot{x}^{L}$.

• •

$(p^{R}_{\eta}\restriction\bar{\bar{\gamma}},\dot{r}_{\eta}\restriction\bar{\bar{\gamma}})\Vdash\check{x}^{R}_{\eta}\leq\dot{x}^{R}$.

• •

$t_{\eta}^{L}$ is the unique element of $p_{\eta}^{L}(0)(\alpha)$.

• •

$t_{\eta}^{R}$ is the unique element of $p_{\eta}^{R}(0)(\alpha)$.

• •

$(p_{\eta}^{L}\restriction M_{\alpha})(0)\ast\check{t}^{L}_{\eta}$, $(p_{\eta}^{R}\restriction M_{\alpha})(0)\ast\check{t}^{R}_{\eta}$ are flat conditions in $\mathbb{P}(\mu,<\alpha)\ast\dot{\mathbb{T}}_{\alpha}$, where $\dot{\mathbb{T}}_{\alpha}=\check{\mathbb{T}}_{\bigcup\dot{G}_{\mathbb{P}(\mu,<\alpha)}}$

Finally, set $p^{L}_{*}=p^{L}_{\vartheta}$, $p^{R}_{*}=p^{R}_{\vartheta}$. Then this pair $(p^{L}_{*},p^{R}_{*})$ together with $\langle r_{\eta},\xi_{\eta},\check{x}^{L}_{\eta},\check{x}^{R}_{\eta}\colon\eta<\vartheta\rangle$ are as desired. ∎

Following [2], let us call the sequence $\langle(r_{\eta},\xi_{\eta},\check{x}^{L}_{\eta},\check{x}^{R}_{\eta})\colon\eta<\vartheta\rangle$ an $\alpha$-separating witness for the nodes $\dot{x}^{L}$, $\dot{x}^{R}$ relative to $p_{*}^{L}$, $p_{*}^{R}$. We now continue with the proof of Lemma 13:

Claim 19**.**

There is $X\in\mathcal{F}$ such that for every condition $p\in\mathbb{I}^{2}_{\gamma}\cap M$ and $\alpha\in X$ there are conditions $p^{L},p^{R}\leq p$ (both in $M$) such that $p^{L}\restriction M_{\alpha}=p^{R}\restriction M_{\alpha}$ and for every $\gamma^{\prime}\in\gamma\cap M_{\alpha}$, any pair of elements above $\alpha$ in $\operatorname{dom}{(p^{L}(\gamma^{\prime}))}\times\operatorname{dom}{(p^{R}(\gamma^{\prime}))}$ has an $\alpha$-separating witness in $M_{\alpha}$ relative to $p^{L}\restriction\gamma$, $p^{R}\restriction\gamma$.

We call such a pair $(p^{L},p^{R})$ an $\alpha$-separating pair.

Proof.

Recall that we chose a sequence $\left\{\gamma_{i}\colon i<\operatorname{cf}{\gamma}\right\}$ cofinal in $\gamma$. For each $i<\operatorname{cf}{\gamma}$ let $X_{\gamma_{i}}$ be as in Claim 17, and let $X=\bigcap_{i<\operatorname{cf}{\gamma}}{X_{\gamma_{i}}}$. This $X$ suffices, as may be seen by applying Claim 17 $\operatorname{cf}{\gamma}$ many times and using the $<\mu$-strategic closure of $\mathbb{I}^{2}_{\gamma}$. ∎

Returning to the proof of the $\kappa$-c.c., let $X$ be as in Claim 19 and let $\langle p_{\alpha}\colon\alpha<\kappa\rangle\in M$ be a sequence of conditions in $\mathbb{I}^{2}_{\gamma}$. For every $\alpha\in X$ we may extend $p_{\alpha}$ to an $\alpha$-separating pair $(p_{\alpha}^{L},p_{\alpha}^{R})\in M$. Let $s_{\alpha}\in M_{\alpha}$ be the list of separating witnesses and let $\bar{p}_{\alpha}$ denote $p_{\alpha}^{L}\restriction M_{\alpha}=p_{\alpha}^{R}\restriction M_{\alpha}$.

The function $\alpha\mapsto(s_{\alpha},\bar{p}_{\alpha})$ is regressive, and so by normality of $\mathcal{F}$ there is a set $Y$ which is positive with respect to this filter and a pair $(s^{*},\bar{p}^{*})$ such that for all $\alpha\in Y$ $(s_{\alpha},\bar{p}_{\alpha})=(s^{*},\bar{p}^{*})$. By further thinning we may assume that for every $\alpha_{0},\alpha_{1}\in Y$ with $\alpha_{0}<\alpha_{1}$ we have $p_{\alpha_{0}}^{L},p_{\alpha_{1}}^{R}\in M_{\alpha_{1}}$. Similarly, we may assume without loss of generality that

[TABLE]

is a $\Delta$-system with root $R$.

We claim that for any $\alpha_{0}<\alpha_{1}$ in $Y$, $p_{\alpha_{0}}$ is compatible with $p_{\alpha_{1}}$, as witnessed by the condition $q$ given by $q(\gamma^{\prime})=p_{\alpha_{0}}^{L}(\gamma^{\prime})\cup p_{\alpha_{1}}^{R}(\gamma^{\prime})$ for every $\gamma^{\prime}<\gamma$.

We must show that $q$ so defined is a condition. Clearly $|\operatorname{dom}{(q)}|<\mu$ and so it remains only to show that $q\restriction\gamma^{\prime}$ forces that $q(\gamma^{\prime})$ is a condition in $\dot{\mathbb{J}}^{2}_{\gamma^{\prime}}$ for all $\gamma^{\prime}<\gamma$. We proceed by induction on $\gamma^{\prime}$. For $\gamma^{\prime}=0$ $q(\gamma^{\prime})\in\mathbb{P}(\mu,<\kappa)$, since $p^{L}_{\alpha_{0}}(\gamma^{\prime})$, $p^{R}_{\alpha_{1}}(\gamma^{\prime})$ have identical intersection with $M_{\alpha_{0}}$ and are disjoint above $\alpha_{0}$.

Now assume $\gamma^{\prime}>0$ and $q\restriction\gamma^{\prime}$ is a condition. Without loss of generality $\dot{T}_{\gamma^{\prime}_{0}}$ is an $\mathbb{I}^{2}_{\gamma^{\prime}}\ast\dot{\mathbb{I}}^{1}_{\gamma^{\prime}}$-name for a $\kappa$-Aronszajn tree, as otherwise $\dot{\mathbb{J}}^{2}_{\gamma^{\prime}}$ is a name for the trivial forcing. We may also assume $\gamma^{\prime}\in R$, since otherwise $q(\gamma^{\prime})$ is either $p^{L}_{\alpha_{0}}(\gamma^{\prime})$ or $p_{\alpha_{1}}^{R}(\gamma^{\prime})$.

In order to show that $q\restriction\gamma^{\prime}\Vdash q(\gamma^{\prime})$ is a condition we must show that if $\dot{x}^{L},\dot{x}^{R}\in\operatorname{dom}{(q(\gamma^{\prime}))}$ and $q\restriction\gamma^{\prime}\Vdash q(\gamma^{\prime})(\dot{x}^{L})=q(\gamma^{\prime})(\dot{x}^{R})$, then

[TABLE]

Since $p_{\alpha_{0}}^{L}\restriction M_{\alpha_{0}}=\bar{p}=p^{R}_{\alpha_{1}}\restriction M_{\alpha_{1}}$, we may assume without loss of generality that both $\dot{x}^{L}$, $\dot{x}^{R}$ are names for nodes above level $\alpha_{0}$. Letting $s^{*}=\langle r_{\eta},\xi_{\eta},\check{x}^{L}_{\eta},\check{x}^{R}_{\eta}\colon\eta<\vartheta\rangle$ be our fixed separating witness, we have:

[TABLE]

By the induction hypothesis $q\restriction\gamma^{\prime}$ is a condition extending both $p_{\alpha_{0}}^{L}\restriction\gamma^{\prime}$ and $p_{\alpha_{0}}^{R}\restriction\gamma^{\prime}$ and so in particular, since $\check{x}^{L}_{\eta}\neq\check{x}^{R}_{\eta}$, for all $\eta<\vartheta$ we have

[TABLE]

Since $\left\{\dot{r}_{\eta}\colon\eta<\vartheta\right\}$ is forced to be a maximal antichain, we have

[TABLE]

as desired. ∎

Remark*.*

It behooves us to observe that the proof of Lemma 13 actually gives us something stronger–namely that for any $S\subseteq\delta$ such that $\mathbb{I}^{2}_{\delta}\restriction S$ is a regular subposet of $\mathbb{I}^{2}_{\delta}$, and for any $\mathbb{I}^{2}_{\delta}\restriction S$-generic $K$, the quotient $\mathbb{I}^{2}_{\delta}/K$ is $\kappa$-Knaster. The proof of this is almost identical to that of Lemma 13, but we must observe that every name $\dot{T}_{\gamma}$ for a $\kappa$-Aronszajn tree considered in the iteration remains a name for a $\kappa$-Aronszajn tree in $V[K]$. This is true since any such tree is $\kappa$-Aronszajn (in fact, special) in $V[K][L]$, where $L$ is generic for $\mathbb{I}^{2}_{\delta}/K$.

From Lemmas 12 and 13 we have

Lemma 20**.**

For all $\gamma\leq\delta$ $\mathbb{I}^{2}_{\gamma}\ast\dot{\mathbb{I}}^{1}_{\gamma}$ has the $\kappa$-c.c.

Lemma 21**.**

Let $\mathbb{I}=\mathbb{I}^{2}_{\delta}\ast\dot{\mathbb{I}}^{1}_{\delta}$ be as above and suppose that $G$ is $\mathbb{I}$-generic over $V$. Then:

1. (a)

$\mu$* remains a cardinal in $V[G]$, $(\mu^{+})^{V[G]}=\kappa$, and $(\mu^{++})^{V[G]}=(\kappa^{+})^{V}$.* 2. (b)

$V[G]\models 2^{\mu}\geq\delta$. 3. (c)

$V[G]\models\square_{\mu,2}$.

Now we are ready to complete the proof of Theorem 10.

Lemma 22**.**

Suppose that $X\in V[G]$ and $X\subseteq\kappa$. Then $X\in V[G_{\mathbb{I}^{2}_{\gamma}\ast\dot{\mathbb{I}}^{1}_{\gamma}}]$ for some $\gamma<\delta$.

Proof.

Immediate from Lemma 20. ∎

Lemma 23**.**

The poset $\mathbb{I}$ forces $\mathsf{SATP}(\kappa)$.

Proof.

Suppose that $T$ is an $\kappa$-Aronszajn tree in $V[G_{\mathbb{I}}]$ and let $\dot{T}$ be a canonical name for it. Then for some $\gamma<\delta$ it is an $\mathbb{I}_{\gamma}$-name and $\dot{T}=\dot{T}_{\gamma}$. By construction of the forcing poset, $\Vdash_{\mathbb{I}_{\gamma+1}}``\dot{T}$ is special” and since $V[G_{\mathbb{I}_{\gamma+1}}]$ and $V[G_{\mathbb{I}}]$ have the same cardinals $T$ remains special in the latter generic extension. ∎

This concludes the proof of Theorem 10.

Corollary 24**.**

Suppose that there is a weakly compact cardinal. Then in some generic extension of $V$ the following holds:

[TABLE]

Proof.

Let $\kappa$ be weakly compact and let

[TABLE]

be two iterations such that $\vec{\mathbb{I}}^{2}$ collapses $\kappa$ to $\aleph_{2}$, adds $\square_{\omega_{1},2}$, and specializes all $\kappa$-Aronszajn trees while anticipating $\dot{\vec{\mathbb{I}}}^{1}$–where $\dot{\vec{\mathbb{I}}}^{1}$ is an $\vec{\mathbb{I}}^{2}$-name for the Baumgartner forcing–i.e. a finite support iteration of posets of finite approximations to specializing functions of trees chosen according to some appropriate bookkeeping function. For each ordinal $\gamma\leq\delta$ let $M$ be an arbitrary elementary substructure of $H(\theta)$ ($\theta$ sufficiently large) of cardinality $\kappa$ such that $V_{\kappa}\cup M^{<\kappa}\cup\left\{\gamma\right\}\subseteq M$ and $M$ contains all relevant parameters, and let $\phi\colon\kappa\rightarrow M$ be an arbitrary bijection. Then $\vec{\mathbb{I}}^{2}$, $\dot{\vec{\mathbb{I}}}^{1}$ are suitable for $M$, $\phi$, $\gamma$ by the remark after the proof of Lemma 13 together with [3]. Therefore the result follows immediately from Theorem 10. ∎

5 The Global Result

Using the methods of [2] we are able to obtain our result simulaneously at $\omega$ many successive cardinals.

Theorem 25**.**

Let $\mu$ be an uncountable regular cardinal and assume that there are $\omega$ many supercompact cardinals above $\mu$. Then there is a generic extension of $V$ in which $\square_{\mu^{+n},2}+\mathsf{SATP}(\mu^{+n+1})$ holds for all $n\geq 0$. Furthermore, if $\mu=\aleph_{1}$, we may ensure that $\mathsf{SATP}(\aleph_{1})$ holds as well.

Proof.

Let $\langle\kappa_{n}\colon n<\omega\rangle$ be an increasing sequence of indestructibly supercompact cardinals above $\mu$ and let $\delta=(\sup_{n<\omega}{\kappa_{n}})^{++}$. In the following it will be convenient to write $\kappa_{-1}=\mu$. If $\mu=\aleph_{1}$, let $R=\omega\cup\left\{-1\right\}$, and otherwise let $R=\omega$. Let $h\colon\delta\setminus\left\{0\right\}\rightarrow R$ be a function such that for all $n\in R$, $h(\alpha)=n$ for unboundedly many $\alpha<\delta$. We define an iteration

[TABLE]

as well as auxilliary forcings $\mathbb{I}_{\gamma}(>\kappa_{n})$, $\dot{\mathbb{I}}_{\gamma}(\kappa_{n})$, and $\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$ for $n\in R$, $\gamma\leq\delta$ such that:

1. (a)

$\mathbb{I}_{\gamma}\simeq\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{I}}_{\gamma}(\kappa_{n})\ast\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$ 2. (b)

$\mathbb{I}_{\gamma}(>\kappa_{n})$ is $<\kappa_{n}$-strategically closed. 3. (c)

For all $n\geq 0$, $\Vdash_{\mathbb{I}_{\gamma}(>\kappa_{n})}\dot{\mathbb{I}}_{\gamma}(\kappa_{n})$ is $\kappa_{n}$-c.c. and $<\kappa_{n-1}$- strategically closed, and for $n=-1$, $\Vdash_{\mathbb{I}_{\gamma}(>\kappa_{n})}\dot{\mathbb{I}}_{\gamma}(\kappa_{n})$ is c.c.c. 4. (d)

For all $n\geq 0$, $\Vdash_{\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{I}}_{\gamma}(\kappa_{n})}\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$ is $\kappa_{n-1}$-c.c. and $<\mu$-closed.

Set $\mathbb{I}_{1}=\prod_{n<\omega}{\mathbb{P}(\kappa_{n-1},<\kappa_{n})}$, where the product is taken with full support, and then let

• •

$\mathbb{I}_{1}(<\kappa_{n})=\prod_{m<n}{\mathbb{P}(\kappa_{m-1},<\kappa_{m})}$ for $n\geq 0$ and is the trivial forcing for $n=-1$.

• •

$\mathbb{I}_{1}(\kappa_{n})=\mathbb{P}(\kappa_{n-1},<\kappa_{n})$ for $n\geq 0$ and is the trivial forcing for $n=-1$.

• •

$\mathbb{I}_{1}(>\kappa_{n})=\prod_{m>n}{\mathbb{P}(\kappa_{m-1},<\kappa_{m})}$, also taken with full support.

Now suppose that $2\leq\gamma\leq\delta$ and we have already defined $\mathbb{I}_{\gamma^{\prime}}$, $\mathbb{I}_{\gamma^{\prime}}(>\kappa_{n})$, $\dot{\mathbb{I}}_{\gamma^{\prime}}(\kappa_{n})$, and $\dot{\mathbb{I}}_{\gamma^{\prime}}(<\kappa_{n})$ for all $\gamma^{\prime}<\gamma$ and $n\in R$. We define $\mathbb{I}_{\gamma}$, $\dot{\mathbb{I}}_{\gamma}(>\kappa_{n})$, and $\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$ as follows:

• •

If $\gamma$ is a limit ordinal then $\mathbb{I}_{\gamma}$ is the set of all $p$ with domain $\gamma$ such that $p\restriction\gamma^{\prime}\in\mathbb{I}_{\gamma^{\prime}}$ for all $\gamma^{\prime}<\gamma$, for all $n\geq 0$ we have $|\operatorname{supp}{(p)}\cap h^{-1}(n)|<\kappa_{n-1}$, and for $n=-1$ we have $|\operatorname{supp}{(p)}\cap h^{-1}(n)|$ finite.

• •

If $\gamma=\bar{\gamma}+1$ is a successor ordinal and $n=h(\bar{\gamma})$ then let $\dot{T}_{\bar{\gamma}}$ be an $\mathbb{I}_{\bar{\gamma}}$-name for a $\kappa_{n}$-Aronszajn tree chosen according to some bookkeeping function, and let

[TABLE]

Observe that $\dot{\mathbb{B}}_{\mathbb{I}_{\bar{\gamma}}(<\kappa_{n})}(\dot{T}_{\bar{\gamma}})$ is an $\mathbb{I}_{\bar{\gamma}}(>\kappa_{n-1})$-name (if $n=-1$ we mean here that it is an $\mathbb{I}_{\bar{\gamma}}$-name) and so may be viewed as an $\mathbb{I}_{\bar{\gamma}}(\kappa_{n})$-name in the extension by $\mathbb{I}_{\bar{\gamma}}(>\kappa_{n})$.

For $n\in R$ let

[TABLE]

Then $\mathbb{I}_{\gamma}(>\kappa_{n})$ is a regular subforcing of $\mathbb{I}_{\gamma}$. We let $\dot{\mathbb{I}}_{\gamma}(\kappa_{n})$ be an $\mathbb{I}_{\gamma}(>\kappa_{n})$-name for the poset

[TABLE]

and let $\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$ be an $\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{I}}_{\gamma}(\kappa_{n})$-name for the poset

[TABLE]

Observe that $\mathbb{I}_{\gamma}\simeq\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{I}}_{\gamma}(\kappa_{n})\ast\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$.

Lemma 26**.**

Let $G_{>\kappa_{n}}$ be generic for $\mathbb{I}_{\gamma}(>\kappa_{n})$ and $\dot{G}_{\kappa_{n}}$ be an $\mathbb{I}_{\gamma}(>\kappa_{n})$-name for the generic for $\dot{\mathbb{I}}_{\gamma}(\kappa_{n})$. If $n\geq 0$, then $V[G_{>\kappa_{n}}]\models$ “$\mathbb{I}_{\gamma}(\kappa_{n})$ is $<\kappa_{n-1}$-strategically closed” and $V[G_{>\kappa_{n}}\ast\dot{G}_{\kappa_{n}}]\models$ “$\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$ is $<\mu$-strategically closed.”

Proof.

Clearly

[TABLE]

since $\mathbb{I}_{\gamma}(\kappa_{n})$ may be defined as an iteration with $<\kappa_{n-1}$-support in $V[G_{>\kappa_{n}}]$ where each iterand has the requisite closure. The fact that

[TABLE]

may be argued similarly. ∎

Lemma 27**.**

Let $G_{>\kappa_{n}}$, $\dot{G}_{\kappa_{n}}$ be as in the statement of Lemma 26. Then $V\left[G_{>\kappa_{n}}\right]\models$ “$\mathbb{I}_{\gamma}(\kappa_{n})$ is $\kappa_{n}$-Knaster” and $V[G_{>\kappa_{n}}\ast\dot{G}_{\kappa_{n}}]\models$ “$\dot{\mathbb{I}}_{\gamma}(<\kappa_{n})$ is $\kappa_{n-1}$-Knaster.” Moreover, we actually have $V[G_{>\kappa_{n}}]\models$ “$\mathbb{I}_{\gamma}(\kappa_{n})/L$ is $\kappa_{n}$-Knaster,” for any $L$ which is generic for a regular subiteration of $\mathbb{I}_{\gamma}(\kappa_{n})$.

Proof.

We prove these simultaneously using induction on $n$. For each $m>n$ let $G_{\kappa_{m}}$ denote the generic for $\mathbb{I}_{\gamma}(\kappa_{m})$ and let $\dot{\mathbb{T}}(\kappa_{m})=\check{\mathbb{T}}_{\bigcup\dot{G}_{\kappa_{m}}}$ be the $\mathbb{I}_{\gamma}(\kappa_{m})$-name for the poset which threads $\bigcup G_{\kappa_{m}}$ with conditions of size $<\kappa_{m-1}$. An argument similar to that given in Lemma 4 tells us that $\mathbb{I}_{\gamma}(\kappa_{m})\ast\dot{\mathbb{T}}(\kappa_{m})$ is $<\kappa_{m-1}$-closed. Moreover, it is clear that this poset forces $\lvert\kappa_{m}\rvert=\kappa_{m-1}$ and has size $\leq\delta$. Let $\dot{\mathbb{T}}(>\kappa_{n})$ be the $\mathbb{I}_{\gamma}(>\kappa_{n})$-name for

[TABLE]

where $\mathbb{T}(\kappa_{m})=\dot{\mathbb{T}}(\kappa_{m})\left[G_{\kappa_{m}}\right]$ and the product is taken with full support. Then $\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{T}}(>\kappa_{n})$ is $<\kappa_{n}$-closed, forces $\lvert\kappa_{m}\rvert=\kappa_{n}$ for all $m>n$, and has cardinality $\leq\delta$, and so there is a regular embedding from $\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{T}}(>\kappa_{n})$ into $\operatorname{Col}{(\kappa_{n},\delta)}$–in fact we have $\operatorname{Col}{(\kappa_{n},\delta)}\simeq(\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{T}}(>\kappa_{n}))\times\operatorname{Col}{(\kappa_{n},\delta)}$. Let $(G_{>\kappa_{n}}\ast\dot{H}_{>\kappa_{n}})\times K_{n}$ be generic for the latter poset.

Then to show that $\mathbb{I}_{\gamma}(\kappa_{n})$ is $\kappa_{n}$-Knaster in $V[G_{>\kappa_{n}}]$ it suffices to show that it satisfies this property in $V[(G_{>\kappa_{n}}\ast\dot{H}_{>\kappa_{n}})\times K_{n}]$. But $(\mathbb{I}_{\gamma}(>\kappa_{n})\ast\dot{\mathbb{T}}(>\kappa_{n}))\times\operatorname{Col}{(\kappa_{n},\delta)}\simeq\operatorname{Col}{(\kappa_{n},\delta)}$ is $<\kappa_{n}$-directed closed and therefore $\kappa_{n}$ is supercompact (and in particular weakly compact) in $V[(G_{>\kappa_{n}}\ast\dot{H}_{>\kappa_{n}})\times K_{n}]$. Thus $\mathbb{I}_{\gamma}(\kappa_{n})$ is $\kappa_{n}$-Knaster in this generic extension by Lemma 13 from the proof of Theorem 10. More precisely, we apply Lemma 13 to the pair $\vec{\mathbb{I}}_{\gamma}(\kappa_{n})$, $\dot{\vec{\mathbb{I}}}_{\gamma}(<\kappa_{n})$. Note that in order to do so we must have that this pair is suitable (with regard to sufficiently closed elementary substructures of $H(\theta)$) in the sense of Definition 8. But part (1) of this definition follows from the inductive hypothesis of the current lemma and part (2) follows from Lemma 9 in conjunction with the inductive hypothesis.

For the “moreover” part of the lemma, use the remark after the proof of Lemma 13 rather than Lemma 13 itself.

Finally, for the second part of the lemma, recall that by the inductive hypothesis we have

[TABLE]

Since $G_{>\kappa_{n-1}}=G_{>\kappa_{n}}\ast\,\dot{G}_{\kappa_{n}}$, $\mathbb{I}_{\gamma}(<\kappa_{n})\simeq\mathbb{I}_{\gamma}(\kappa_{n-1})\,\ast\,\dot{\mathbb{I}}_{\gamma}(<\kappa_{n-1})$, and $V[G_{>\kappa_{n-1}}]\models$ “$\dot{\mathbb{I}}_{\gamma}(\kappa_{n-1})$ is $\kappa_{n-1}$-Knaster” by the inductive hypothesis, we have

[TABLE]

as desired. ∎

Now Theorem 25 follows immediately from Lemmas 26 and 27.

∎

Finally we may use an Easton-support iteration of the $\omega$-block posets given by Theorem 25 exactly as in [2] to obtain a global result:

Theorem 28**.**

Assume that there are class many supercompact cardinals with no inaccessible limit. Then there is a class forcing extension of $V$ in which $\square_{\kappa,2}+\mathsf{SATP}(\kappa^{+})$ holds for all regular $\kappa$.