A note on the eightfold way

TL;DR

This paper constructs a set-theoretic model assuming a Mahlo cardinal where certain combinatorial properties of $ ext{cof}( ext{omega})$ subsets of $ ext{omega}_2$ are demonstrated, including the existence of an $ ext{omega}_2$-Aronszajn tree.

Contribution

It introduces a new model under large cardinal assumptions where specific reflection and approachability properties fail or hold, advancing understanding of set-theoretic combinatorics.

Findings

Existence of an $ ext{omega}_2$-Aronszajn tree in the model

Failure of the $ ext{omega}_1$-approachability property

Stationary subsets of $ ext{omega}_2 ext{∩ cof}( ext{omega})$ reflect

Abstract

Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $\omega_2$-Aronszajn tree, the $\omega_1$-approachability property fails, and every stationary subset of $\omega_2 \cap \mathrm{cof}(\omega)$ reflects.

Equations

$$⟨{\mathbb{P}}_{\alpha },{\dot{\mathbb{Q}}}_{\beta }:\alpha \le \kappa ,\beta <\kappa ⟩.$$

$${\mathbb{P}}_{\alpha }\ast \text{Add}(\alpha )\ast \text{Add}(\omega ).$$

$$s_{\alpha }:=P_{{\omega }_1}(\alpha )\setminus V^{{\mathbb{P}}_{\alpha +1}}$$

$$⟨{\mathbb{A}}_i,{\dot{T}}_j:i\le \alpha ,j<\alpha ⟩,$$

$$⟨{\mathbb{A}}_i,{\dot{T}}_j:i\le {\omega }_3,j<{\omega }_3⟩,$$

$$⟨{\mathbb{A}}_i^{\ast },{\dot{T}}_j^{\ast }:i\le \pi (\alpha ),j<\pi (\alpha )⟩:=\pi (⟨{\mathbb{A}}_i,{\dot{T}}_j:i\le \alpha ,j<\alpha ⟩)$$

$$a_{\alpha }:=s_{\alpha }\upharpoonright (\alpha \times \omega )\text{and}{b}_{\alpha }:=s_{\alpha }\upharpoonright ([\alpha ,{\omega }_2)\times \omega ).$$

Full text

2010 Mathematics Subject Classification: Primary 03E35; Secondary 03E05.

Key words and phrases. Stationary reflection, Aronszajn tree, approachability property, disjoint stationary sequence.

The second author was partially supported by the National Science Foundation Grant No. DMS-1464859.

Cummings, Friedman, Magidor, Rinot, and Sinapova [1] proved the consistency of any logical Boolean combination of the statements which assert the $\omega_{1}$-approachability property, the tree property on $\omega_{2}$, and stationary reflection at $\omega_{2}$. For most of these combinations, they assumed the existence of a weakly compact cardinal in order to construct the desired model. This is a natural assumption to make, since the $\omega_{2}$-tree property implies that $\omega_{2}$ is weakly compact in $L$. On the other hand, Harrington and Shelah [4] proved that stationary reflection at $\omega_{2}$ is equiconsistent with the existence of a Mahlo cardinal. Cummings et al. [1] asked whether a Mahlo cardinal is sufficient to prove the consistency of the existence of an $\omega_{2}$-Aronszajn tree, the failure of the $\omega_{1}$-approachability property, and stationary reflection at $\omega_{2}$. In this article we answer this question in the affirmative.

We begin by reviewing the relevant definitions and facts. We refer the reader to [1] for a more detailed discussion of these ideas and their history. A stationary set $S\subseteq\omega_{2}\cap\mathrm{cof}(\omega)$ is said to reflect at an ordinal $\beta\in\omega_{2}\cap\mathrm{cof}(\omega_{1})$ if $S\cap\beta$ is a stationary subset of $\beta$. If $S$ does not reflect at any such ordinal, $S$ is non-reflecting. We say that stationary reflection holds at $\omega_{2}$ if every stationary subset of $\omega_{2}\cap\mathrm{cof}(\omega)$ reflects to some ordinal in $\omega_{2}\cap\mathrm{cof}(\omega_{1})$.

An $\omega_{2}$-Aronszajn tree is a tree of height $\omega_{2}$, whose levels have size less than $\omega_{2}$, and which has no cofinal branches. The $\omega_{2}$-tree property is the statement that there does not exist an $\omega_{2}$-Aronszajn tree. A well-known fact is that if the $\omega_{2}$-tree property holds, then $\omega_{2}$ is a weakly compact cardinal in $L$. Therefore, if one starts with a Mahlo cardinal $\kappa$ which is not weakly compact in $L$ (for example, if $\kappa$ is the least Mahlo cardinal in $L$), then in any subsequent forcing extension in which $\kappa$ equals $\omega_{2}$, there exists an $\omega_{2}$-Aronszajn tree.

The $\omega_{1}$-approachability property is the statement that there exists a sequence $\vec{a}=\langle a_{i}:i<\omega_{2}\rangle$ of countable subsets of $\omega_{2}$ and a club $C\subseteq\omega_{2}$ such that for all limit ordinals $\alpha\in C$, $\alpha$ is approachable by $\vec{a}$ in the following sense: there exists a cofinal set $c\subseteq\alpha$ with order type equal to $\mathrm{cf}(\alpha)$ such that for all $\beta<\alpha$, $c\cap\beta$ is a member of $\{a_{i}:i<\alpha\}$. Essentially, this property is a very weak form of the square principle $\Box_{\omega_{1}}$. The failure of the $\omega_{1}$-approachability property is known to hold in Mitchell’s model [6] in which there does not exist a special $\omega_{2}$-Aronszajn tree, which he constructed using a Mahlo cardinal.

A solution to the problem of [1] addressed in this article was originally discovered by the first author, using a mixed support forcing iteration similar to the forcings appearing in [1] and [2]. Later, the second author found a different proof using the idea of a disjoint stationary sequence. The latter proof is somewhat easier, since it avoids the technicalities of mixed support iterations, and also can be easily adapted to arbitrarily large continuum. In this article we present the second proof.

In Section 1, we discuss the idea of a disjoint stationary sequence, which was originally introduced by the second author in [5]. In Section 2, we prove the main result of the paper. In Section 3, we adapt our model to arbitrarily large continuum using an argument of I. Neeman, which we include with his kind permission.

1. Disjoint Stationary Sequences

Recall that for an uncountable ordinal $\alpha\in\omega_{2}$, $P_{\omega_{1}}(\alpha)$ denotes the set of all countable subsets of $\alpha$. A set $c\subseteq P_{\omega_{1}}(\alpha)$ is club if it is cofinal in $P_{\omega_{1}}(\alpha)$ and closed under unions of countable increasing sequences. A set $s\subseteq P_{\omega_{1}}(\alpha)$ is stationary if it has non-empty intersection with every club in $P_{\omega_{1}}(\alpha)$. For an infinite cardinal $\kappa$, a forcing $\mathbb{P}$ is said to be $\kappa$-distributive if it adds no new subsets of $V$ of size less than $\kappa$.

Let $\alpha$ be an uncountable ordinal in $\omega_{2}$. Fix an increasing and continuous sequence $\langle b_{i}:i<\omega_{1}\rangle$ of countable sets with union equal to $\alpha$ (for example, fix a bijection $f:\omega_{1}\to\alpha$ and let $b_{i}:=f[i]$). Note that the set $\{b_{i}:i<\omega_{1}\}$ is club in $P_{\omega_{1}}(\alpha)$. A set $s\subseteq P_{\omega_{1}}(\alpha)$ is stationary in $P_{\omega_{1}}(\alpha)$ iff the set $x:=\{i<\omega_{1}:b_{i}\in s\}$ is a stationary subset of $\omega_{1}$. Indeed, if $C\subseteq\omega_{1}$ is a club which is disjoint from $x$, then the set $\{b_{i}:i\in C\}$ is a club subset of $P_{\omega_{1}}(\alpha)$ which is obviously disjoint from $s$. On the other hand, if $c\subseteq P_{\omega_{1}}(\alpha)$ is a club which is disjoint from $s$, then the set $\{i<\omega_{1}:b_{i}\in c\}$ is a club in $\omega_{1}$, and this club is clearly disjoint from $x$.

Definition 1.1**.**

A disjoint stationary sequence on $\omega_{2}$ is a sequence $\langle s_{\alpha}:\alpha\in S\rangle$, where $S$ is a stationary subset of $\omega_{2}\cap\mathrm{cof}(\omega_{1})$, satisfying:

1. (1)

for all $\alpha\in S$, $s_{\alpha}$ is a stationary subset of $P_{\omega_{1}}(\alpha)$; 2. (2)

for all $\alpha<\beta$ in $S$, $s_{\alpha}\cap s_{\beta}=\emptyset$.

As we will show below, the existence of a disjoint stationary sequence $\langle s_{\alpha}:\alpha\in S\rangle$ on $\omega_{2}$ implies the failure of the $\omega_{1}$-approachability property (more specifically, that the set $S$ is not in the approachability ideal $I[\omega_{2}]$). In our main result, the failure of the $\omega_{1}$-approachability property will follow from the existence of a disjoint stationary sequence.

One of the advantages of disjoint stationary sequences over other methods for obtaining the failure of approachability, such as using the $\omega_{1}$-approximation property, is their upward absoluteness.

Lemma 1.2**.**

Suppose that $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence. Let $\mathbb{P}$ be a forcing poset which preserves $\omega_{1}$ and $\omega_{2}$, preserves the stationarity of $S$, and preserves stationary subsets of $\omega_{1}$. Then $\mathbb{P}$ forces that $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence.

The proof is straightforward.

Corollary 1.3**.**

Assume that $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence. Let $\mathbb{P}$ be a forcing poset which is either c.c.c., or $\omega_{2}$-distributive and preserves the stationarity of $S$. Then $\mathbb{P}$ forces that $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence.

The next result describes a well-known consequence of approachability; we include a proof for completeness.

Proposition 1.4**.**

Assume that the $\omega_{1}$-approachability property holds. Then for any stationary set $S\subseteq\omega_{2}\cap\mathrm{cof}(\omega_{1})$, there exists an $\omega_{2}$-distributive forcing which adds a club subset of $S\cup(\omega_{2}\cap\mathrm{cof}(\omega))$.

Proof.

Fix a sequence $\vec{a}=\langle a_{i}:i<\omega_{2}\rangle$ of countable subsets of $\omega_{2}$ and a club $C\subseteq\omega_{2}$ such that for all limit ordinals $\alpha\in C$, there exists a set $e\subseteq\alpha$ which is cofinal in $\alpha$, has order type $\mathrm{cf}(\alpha)$, and for all $\beta<\alpha$, $e\cap\beta\in\{a_{i}:i<\alpha\}$.

Define $\mathbb{P}$ as the forcing poset consisting of all closed and bounded subsets of $S\cup(\omega_{2}\cap\mathrm{cof}(\omega))$, ordered by end-extension. We will show that $\mathbb{P}$ is $\omega_{2}$-distributive. Observe that if $c\in\mathbb{P}$ and $\gamma<\omega_{2}$, then there is $d\leq c$ with $\sup(d)\geq\gamma$ (for example, $d:=c\cup\min(S\setminus\max\{\sup(c),\gamma\})$). Using this, a straightforward argument shows that, if $\mathbb{P}$ is $\omega_{2}$-distributive, then $\mathbb{P}$ adds a club subset of $S\cup(\omega_{2}\cap\mathrm{cof}(\omega))$.

To show that $\mathbb{P}$ is $\omega_{2}$-distributive, fix $c\in\mathbb{P}$ and a family $\{D_{i}:i<\omega_{1}\}$ of dense open subsets of $\mathbb{P}$. We will find $d\leq c$ in $\bigcap\{D_{i}:i<\omega_{1}\}$.

Fix a regular cardinal $\theta$ large enough so that all of the objects mentioned so far are members of $H(\theta)$. Fix a well-ordering $\unlhd$ of $H(\theta)$. Since $S$ is stationary, we can find an elementary substructure $N$ of $(H(\theta),\in,\unlhd)$ such that $\vec{a}$, $C$, $S$, $\mathbb{P}$, $c$, and $\langle D_{i}:i<\omega_{1}\rangle$ are members of $N$ and $\alpha:=N\cap\omega_{2}\in S$. In particular, $\alpha\in C\cap\mathrm{cof}(\omega_{1})$. Fix a cofinal set $e\subseteq\alpha$ with order type $\omega_{1}$ such that for all $\beta<\alpha$, $e\cap\beta\in\{a_{i}:i<\alpha\}$. Enumerate $e$ in increasing order as $\langle\gamma_{i}:i<\omega_{1}\rangle$. Note that since $\{a_{i}:i<\alpha\}$ is a subset of $N$ by elementarity, for all $\beta<\alpha$, $e\cap\beta\in N$. Consequently, for each $\delta<\omega_{1}$, the sequence $\langle\gamma_{i}:i<\delta\rangle$ is a member of $N$.

We define by induction a strictly descending sequence of conditions $\langle c_{i}:i<\omega_{1}\rangle$, starting with $c_{0}:=c$, together with some auxiliary objects. We will maintain that for each $\delta<\omega_{1}$, the sequence $\langle c_{i}:i<\delta\rangle$ is definable in $H(\theta)$ from parameters in $N$, and hence is a member of $N$.

Given a limit ordinal $\delta<\omega_{1}$, assuming that $c_{i}$ is defined for all $i<\delta$, we define $c_{\delta,0}$ to be equal to $\bigcup\{c_{i}:i<\delta\}$. Then clearly $\sup(c_{\delta,0})$ is an ordinal of cofinality $\omega$. Hence, $c_{\delta}:=c_{\delta,0}\cup\{\sup(c_{\delta,0})\}$ is a condition and is a strict end-extension of $c_{i}$ for all $i<\delta$. Now assume that $\xi<\omega_{1}$ and $c_{i}$ is defined for all $i\leq\xi$. Let $c_{\xi,0}$ be the $\unlhd$-least strict end-extension of $c_{\xi}$ such that $\max(c_{\xi,0})\geq\gamma_{\xi}$. Now let $c_{\xi+1}$ be the $\unlhd$-least condition in $D_{\xi}$ which is below $c_{\xi,0}$. This completes the construction. Define $d_{0}:=\bigcup\{c_{i}:i<\omega_{1}\}$.

Reviewing the inductive definition of the sequence $\langle c_{i}:i<\omega_{1}\rangle$, we see that for all $\delta<\omega_{1}$, $\langle c_{i}:i<\delta\rangle$ is definable in $H(\theta)$ from parameters in $N$, including specifically the sequence $\langle\gamma_{i}:i<\delta\rangle$. Therefore, each $c_{i}$ is in $N$. In addition, for each $i<\omega_{1}$, $\max(c_{i+1})\geq\gamma_{i}$. Since $\{\gamma_{i}:i<\omega_{1}\}=e$ is cofinal in $\alpha$, $\sup(d_{0})=\alpha$. Let $d:=d_{0}\cup\{\alpha\}$. Then $d$ is a condition since $\alpha\in S$, and $d\leq c_{i}$ for all $i<\omega_{1}$, and in particular, $d\leq c$. For each $i<\omega_{1}$, $c_{i+1}\in D_{i}$, so $d\in D_{i}$. ∎

Proposition 1.5**.**

Suppose that $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence. Then $(\omega_{2}\cap\mathrm{cof}(\omega_{1}))\setminus S$ is stationary.

Proof.

Let $C$ be club in $\omega_{2}$. By induction, it is easy to define an increasing and continuous sequence $\langle N_{i}:i<\omega_{1}\rangle$ satisfying:

1. (1)

each $N_{i}$ is a countable elementary substructure of $H(\omega_{3})$ containing the objects $\langle s_{\alpha}:\alpha\in S\rangle$ and $C$; 2. (2)

for each $i<\omega_{1}$, $N_{i}\in N_{i+1}$.

Let $N:=\bigcup\{N_{i}:i<\omega_{1}\}$. Then by elementarity, $\omega_{1}\subseteq N$ and $\beta:=N\cap\omega_{2}$ has cofinality $\omega_{1}$ and is in $C$.

We claim that $\beta\notin S$, which completes the proof. Suppose for a contradiction that $\beta\in S$. Then $s_{\beta}$ is defined and is a stationary subset of $P_{\omega_{1}}(\beta)$. On the other hand, $\langle N_{i}\cap\omega_{2}:i<\omega_{1}\rangle$ is a club subset of $P_{\omega_{1}}(\beta)$. So we can fix $i<\omega_{1}$ such that $N_{i}\cap\omega_{2}\in s_{\beta}$.

Now the sequence $\langle s_{\alpha}:\alpha\in S\rangle$ is a member of $N$, and also $N_{i}\cap\omega_{2}\in N\cap s_{\beta}$. So by elementarity, there exists $\alpha\in N\cap S$ such that $N_{i}\cap\omega_{2}\in s_{\alpha}$. Then $\alpha\in N\cap\omega_{2}=\beta$, so $\alpha<\beta$. Thus, we have that $N_{i}\cap\omega_{2}$ is a member of both $s_{\alpha}$ and $s_{\beta}$, which contradicts that $s_{\alpha}\cap s_{\beta}=\emptyset$. ∎

Corollary 1.6**.**

Assume that there exists a disjoint stationary sequence on $\omega_{2}$. Then the $\omega_{1}$-approachability property fails.

Proof.

Suppose for a contradiction that $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence and the $\omega_{1}$-approachability property holds. By Proposition 1.4, fix an $\omega_{2}$-distributive forcing $\mathbb{P}$ which adds a club subset of $S\cup(\omega_{2}\cap\mathrm{cof}(\omega))$. In particular, $\mathbb{P}$ forces that $(\omega_{2}\cap\mathrm{cof}(\omega_{1}))\setminus S$ is non-stationary in $\omega_{2}$. By Proposition 1.5, the sequence $\langle s_{\alpha}:\alpha\in S\rangle$ is not a disjoint stationary sequence in $V^{\mathbb{P}}$.

Now $\mathbb{P}$ is $\omega_{2}$-distributive, and it preserves the stationarity of $S$ because it adds a club subset of $S\cup(\omega_{2}\cap\mathrm{cof}(\omega))$. By Corollary 1.3, $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence in $V^{\mathbb{P}}$, which is a contradiction. ∎

2. The main result

Assume for the rest of the section that $\kappa$ is a Mahlo cardinal. Without loss of generality, we may also assume that $2^{\kappa}=\kappa^{+}$, since this can be forced while preserving Mahloness. Define $S$ as the set of inaccessible cardinals below $\kappa$.

We will define a two-step forcing iteration $\mathbb{P}*\dot{\mathbb{A}}$ with the following properties. The forcing $\mathbb{P}$ collapses $\kappa$ to become $\omega_{2}$ and adds a disjoint stationary sequence on $S$. In $V^{\mathbb{P}}$, $\mathbb{A}$ is an iteration for destroying the stationarity of non-reflecting subsets of $\kappa\cap\mathrm{cof}(\omega)$. The forcing $\mathbb{A}$ will be $\kappa$-distributive and preserve the stationarity of $S$, which implies by Corollary 1.3 that there exists a disjoint stationary sequence in $V^{\mathbb{P}*\dot{\mathbb{A}}}$. Thus, in $V^{\mathbb{P}*\dot{\mathbb{A}}}$ we have that stationary reflection holds at $\omega_{2}$ and the $\omega_{1}$-approachability property fails. If, in addition, we assume that the Mahlo cardinal $\kappa$ is not weakly compact in $L$, then there exists an $\omega_{2}$-Aronszajn tree in $V^{\mathbb{P}*\dot{\mathbb{A}}}$ as discussed above.

The remainder of this section is divided into two parts. In the first part we will develop the forcing $\mathbb{P}$, and in the second we will handle the forcing $\mathbb{A}$ in $V^{\mathbb{P}}$. We will use the following theorem of Gitik [3]. Suppose that $V\subseteq W$ are transitive models of ZFC with the same ordinals and the same $\omega_{1}$ and $\omega_{2}$. If $(P(\omega)\cap W)\setminus V$ is non-empty, then in $W$ the set $P_{\omega_{1}}(\omega_{2})\setminus V$ is stationary in $P_{\omega_{1}}(\omega_{2})$. For a regular cardinal $\kappa$, we let $\textrm{Add}(\kappa)$ denote the usual Cohen forcing consisting of all functions from some $\gamma<\kappa$ into $2$, ordered by reverse inclusion.

We define by induction a forcing iteration

[TABLE]

This iteration will be a countable support forcing iteration of proper forcings. We will then let $\mathbb{P}:=\mathbb{P}_{\kappa}$.

Fix $\alpha<\kappa$ and assume that $\mathbb{P}_{\alpha}$ has been defined. We split the definition of $\dot{\mathbb{Q}}_{\alpha}$ into three cases. If $\alpha$ is an inaccessible cardinal, then let $\dot{\mathbb{Q}}_{\alpha}$ be a $\mathbb{P}_{\alpha}$-name for the forcing $\textrm{Add}(\alpha)$. If $\alpha=\beta+1$ where $\beta$ is inaccessible, then let $\dot{\mathbb{Q}}_{\alpha}$ be a $\mathbb{P}_{\alpha}$-name for $\textrm{Add}(\omega)$. For all other cases, let $\dot{\mathbb{Q}}_{\alpha}$ be a $\mathbb{P}_{\alpha}$-name for $\mathrm{Col}(\omega_{1},\omega_{2})$. Note that in any case, $\dot{\mathbb{Q}}_{\alpha}$ is forced to be proper. Now let $\mathbb{P}_{\alpha+1}$ be $\mathbb{P}_{\alpha}*\dot{\mathbb{Q}}_{\alpha}$. At limit stages $\delta\leq\kappa$, assuming that $\mathbb{P}_{\alpha}$ is defined for all $\alpha<\delta$, we let $\mathbb{P}_{\delta}$ denote the countable support limit of these forcings.

This completes the construction. For each $\alpha\leq\kappa$, $\mathbb{P}_{\alpha}$ is a countable support iteration of proper forcings, and hence is proper. Also, by standard facts, if $\beta<\alpha$, then $\mathbb{P}_{\beta}$ is a regular suborder of $\mathbb{P}_{\alpha}$, and in $V^{\mathbb{P}_{\beta}}$, the quotient forcing $\mathbb{P}_{\alpha}/\dot{G}_{\mathbb{P}_{\beta}}$ is forcing equivalent to a countable support iteration of proper forcings, and hence is itself proper. We let $\dot{\mathbb{P}}_{\beta,\alpha}$ be a $\mathbb{P}_{\beta}$-name for this proper forcing iteration which is equivalent to $\mathbb{P}_{\alpha}/\dot{G}_{\mathbb{P}_{\beta}}$ in $V^{\mathbb{P}_{\beta}}$.

One can show by well-known arguments that for all inaccessible cardinals $\alpha\leq\kappa$, $\mathbb{P}_{\alpha}$ has size $\alpha$, is $\alpha$-c.c., and forces that $\alpha=\omega_{2}$. Namely, since $\alpha$ is inaccessible, for all $\beta<\alpha$, $|\mathbb{P}_{\beta}|<\alpha$. Hence $\mathbb{P}_{\alpha}$ has size $\alpha$ by definition. A standard $\Delta$-system argument shows that $\mathbb{P}_{\alpha}$ is $\alpha$-c.c., and since collapses are used at cofinally many stages below $\alpha$, $\mathbb{P}_{\alpha}$ turns $\alpha$ into $\omega_{2}$.

Let $\mathbb{P}:=\mathbb{P}_{\kappa}$. In $V^{\mathbb{P}}$, let us define a disjoint stationary sequence. Recall that $S$ is the set of inaccessible cardinals in $\kappa$ in the ground model $V$. Since $\kappa$ is Mahlo, $S$ is a stationary subset of $\kappa$ in $V$. As $\mathbb{P}$ is $\kappa$-c.c., $S$ remains stationary in $V^{\mathbb{P}}$. And since $\mathbb{P}$ is proper and forces that $\kappa=\omega_{2}$, each member of $S$ has cofinality $\omega_{1}$ in $V^{\mathbb{P}}$.

The set $S$ will be the domain of the disjoint stationary sequence in $V^{\mathbb{P}}$. Consider $\alpha\in S$. Then $\mathbb{P}_{\alpha}$ forces that $\alpha=\omega_{2}$. We have that $\mathbb{P}_{\alpha+1}$ is forcing equivalent to $\mathbb{P}_{\alpha}*\textrm{Add}(\alpha)$ and $\mathbb{P}_{\alpha+2}$ is forcing equivalent to

[TABLE]

Clearly, $\alpha$ is still equal to $\omega_{2}$ after forcing with $\mathbb{P}_{\alpha+1}$ or $\mathbb{P}_{\alpha+2}$.

Since there exists a subset of $\omega$ in $V^{\mathbb{P}_{\alpha+2}}\setminus V^{\mathbb{P}_{\alpha+1}}$, in $V^{\mathbb{P}_{\alpha+2}}$ the set

[TABLE]

is a stationary subset of $P_{\omega_{1}}(\alpha)$ by Gitik’s theorem. Now the tail of the iteration $\mathbb{P}_{\alpha+2,\kappa}$ is proper in $V^{\mathbb{P}_{\alpha+2}}$. Therefore, $s_{\alpha}$ remains stationary in $P_{\omega_{1}}(\alpha)$ in $V^{\mathbb{P}}$.

Observe that if $\alpha<\beta$ are both in $S$, then by definition $s_{\alpha}\subseteq V^{\mathbb{P}_{\alpha+2}}\subseteq V^{\mathbb{P}_{\beta}}$, whereas $s_{\beta}\cap V^{\mathbb{P}_{\beta}}=\emptyset$. Thus, $s_{\alpha}\cap s_{\beta}=\emptyset$. It follows that in $V^{\mathbb{P}}$, $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence on $\omega_{2}$.

For the second part of our proof, we work in $V^{\mathbb{P}}$ to define a forcing iteration $\mathbb{A}$ of length $\kappa^{+}$ which is designed to destroy the stationarity of any subset of $\omega_{2}\cap\mathrm{cof}(\omega)$ which does not reflect to an ordinal in $\omega_{2}\cap\mathrm{cof}(\omega_{1})$. This forcing will be shown to be $\kappa$-distributive and preserve the stationarity of $S$. It follows from Corollary 1.3 that $\mathbb{A}$ preserves the fact that $\langle s_{\alpha}:\alpha\in S\rangle$ is a disjoint stationary sequence. Note that since $\mathbb{P}$ is $\kappa$-c.c. and has size $\kappa$, easily $2^{\kappa}=\kappa^{+}$ in $V^{\mathbb{P}}$.

The definition of and arguments involving $\mathbb{A}$ are essentially the same as in the original construction of Harrington and Shelah [4]. The main differences are that we are using $\mathbb{P}$ to collapse $\kappa$ to become $\omega_{2}$ instead of $\mathrm{Col}(\omega_{1},<\!\kappa)$, and that we are now required to show that $\mathbb{A}$ preserves the stationarity of $S$. We will sketch the main points of the construction, but leave some of the routine technical details to be checked by the reader in consultation with [4].

Many of the facts which we will need to know about $\mathbb{A}$ can be abstracted out more generally to a kind of forcing iteration which we will call a suitable iteration. So before defining $\mathbb{A}$, let us describe this kind of iteration in detail. We will assume in what follows that $2^{\omega_{1}}=\omega_{2}$.

Let us define abstractly the idea of a suitable iteration

[TABLE]

where $\alpha\leq\omega_{3}$. Such an iteration is determined by the following recursion. A condition in $\mathbb{A}_{i}$ is any function $p$ whose domain is a subset of $i$ of size less than $\omega_{2}$ such that for all $j\in\mathrm{dom}(p)$, $p(j)$ is a non-empty closed and bounded subset of $\omega_{2}$ such that $p\upharpoonright j$ forces in $\mathbb{A}_{j}$ that $p(j)\cap\dot{T}_{j}=\emptyset$. We let $q\leq p$ if $\mathrm{dom}(p)\subseteq\mathrm{dom}(q)$ and for all $i\in\mathrm{dom}(p)$, $q(i)$ is an end-extension of $p(i)$. And $\dot{T}_{i}$ is a nice $\mathbb{A}_{i}$-name for a subset of $\omega_{2}\cap\mathrm{cof}(\omega)$.111In our construction below, our specific suitable iteration will be shown to be $\omega_{2}$-distributive. However, being $\omega_{2}$-distributive is not a part of the abstract definition of a suitable iteration.

Suppose that $M$ is a transitive model of $\textsf{ZFC}^{-}$ which is closed under $\omega_{1}$-sequences. Then if $M$ models that $\langle\mathbb{A}_{i},\dot{T}_{j}:i\leq\alpha,j<\alpha\rangle$ is a suitable iteration, then in fact it is. Specifically, all the notions used in the recursion above are upwards absolute for such a model, since $M$ contains all $\omega_{1}$-sized sets. For example, $M$ contains all closed and bounded subsets of $\omega_{2}$ and being a nice name is absolute.

Observe that if $\alpha<\omega_{3}$, then $2^{\omega_{1}}=\omega_{2}$ immediately implies that $\mathbb{A}_{\alpha}$ has size $\omega_{2}$. On the other hand, if $\alpha=\omega_{3}$, then a straightforward application of the $\Delta$-system lemma shows that $\mathbb{A}_{\omega_{3}}$ is $\omega_{3}$-c.c. Using a covering and nice name argument, it then follows that if $\mathbb{A}_{\beta}$ is $\omega_{2}$-distributive for all $\beta<\omega_{3}$, then so is $\mathbb{A}_{\omega_{3}}$.

Lemma 2.1**.**

Suppose that for all $i<\alpha$, $\mathbb{A}_{i}$ forces that $\dot{T}_{i}$ is non-stationary. Then for any $q\in\mathbb{A}_{\alpha}$, $\mathbb{A}_{\alpha}/q$ is forcing equivalent to $\textrm{Add}(\omega_{2})$.

Proof.

First we claim that $\mathbb{A}_{\alpha}$ contains an $\omega_{2}$-closed dense subset. For each $i$ let $\dot{E}_{i}$ be an $\mathbb{A}_{i}$-name for a club disjoint from $\dot{T}_{i}$. Define $D$ as the set of conditions $p$ such that for all $i\in\mathrm{dom}(p)$, $p\upharpoonright i$ forces that $\max(p(i))\in\dot{E}_{i}$. It is easy to prove that $D$ is dense and $\omega_{2}$-closed.

Reviewing the definition of $\mathbb{A}_{\alpha}$, clearly $\mathbb{A}_{\alpha}$ is separative and every condition in it has $\omega_{2}$-many incompatible extensions. By a well-known fact, any $\omega_{2}$-closed separative forcing of size $\omega_{2}$ for which any condition has $\omega_{2}$-many incompatible extensions is forcing equivalent to $\textrm{Add}(\omega_{2})$. ∎

Having described the main facts which we will use about a suitable iteration, let us show how this kind of iteration can be used to obtain a model satisfying that stationary reflection holds at $\omega_{2}$. Suppose that we have a ground model in which $2^{\omega_{2}}=\omega_{3}$. Using a standard bookkeeping argument, we can define a suitable iteration

[TABLE]

so that every nice name for a non-reflecting subset of $\omega_{2}\cap\mathrm{cof}(\omega)$ is equal to some $\dot{T}_{j}$. Specifically, assuming that $\mathbb{A}_{i}$ is defined for some $i<\omega_{3}$, then using $2^{\omega_{2}}=\omega_{3}$ and the fact that $\mathbb{A}_{i}$ has size $\omega_{2}$, we can list out all nice $\mathbb{A}_{i}$-names for subsets of $\omega_{2}\cap\mathrm{cof}(\omega)$ in order type $\omega_{3}$. Now choose $\dot{T}_{i}$ to be the first name (according to the bookkeeping function) which was listed at some stage less than or equal to $i$ which is forced by $\mathbb{A}_{i}$ to be non-reflecting. In this manner, we can arrange that after $\omega_{3}$-many stages, all names which arise during the iteration are handled, and thus that the iteration destroys the stationarity of all non-reflecting sets. Of course this construction breaks down if we reach some $i$ such that $\mathbb{A}_{i}$ is not $\omega_{2}$-distributive. So proving the $\omega_{2}$-distributivity of such a suitable iteration will be the main remaining goal.

This completes the abstract description of a suitable iteration and how it will be used to obtain stationary reflection at $\omega_{2}$. Let us now return to our construction. Fix a generic filter $G$ on $\mathbb{P}$. Then in $V[G]$ we have that $\kappa=\omega_{2}$, $2^{\omega_{1}}=\omega_{2}$, and $2^{\omega_{2}}=\omega_{3}=\kappa^{+}$. Working in $V[G]$, we define a suitable iteration $\langle\mathbb{A}_{i},\dot{T}_{j}:i\leq\kappa^{+},j<\kappa^{+}\rangle$. We will prove that each $\mathbb{A}_{i}$ is $\omega_{2}$-distributive and preserves the stationarity of $S$. By the discussion above, this will complete the proof of our main result.

Fix $\alpha<\kappa^{+}$. In $V$, fix $\mathbb{P}$-names $\dot{\mathbb{A}}_{i}$ for all $i\leq\alpha$ and $\dot{T}_{j}$ for all $j<\alpha$ which are forced to satisfy the definitions of these objects given above (we will abuse notation by writing $\dot{T}_{j}$ for the $\mathbb{P}$-name for the $\mathbb{A}_{j}$-name $\dot{T}_{j}$).

We would like to prove that $\mathbb{A}_{\alpha}$ is $\kappa$-distributive and preserves the stationarity of $S$. In order to prove this, we will make two inductive hypotheses. The first inductive hypothesis is that for all $\beta<\alpha$, $\mathbb{A}_{\beta}$ is $\kappa$-distributive and preserves the stationarity of $S$.

Before describing the second inductive hypothesis, we need to develop some ideas and notation. For each $\beta\leq\alpha$, define in $V$ the set $\mathcal{X}_{\beta}$ to consist of all sets $N$ satisfying:

1. (1)

$N\prec H(\kappa^{+})$; 2. (2)

$N$ contains as members $\mathbb{P}$ and $\langle\dot{\mathbb{A}}_{i},\dot{T}_{j}:i\leq\beta,j<\beta\rangle$; 3. (3)

$\kappa_{N}:=|N|=N\cap\kappa$ and $N^{<\kappa_{N}}\subseteq N$; 4. (4)

$\kappa_{N}\in S$.

An easy application of the stationarity of $S$ and the inaccessibility of $\kappa$ shows that each $\mathcal{X}_{\beta}$ is a stationary subset of $P_{\kappa}(H(\kappa^{+}))$. Also note that if $N\in\mathcal{X}_{\beta}$ and $\gamma\in N\cap\beta$, then $N\in\mathcal{X}_{\gamma}$.

Consider $N$ in $\mathcal{X}_{\alpha}$. Since $\mathbb{P}$ is $\kappa$-c.c., the maximal condition in $\mathbb{P}$ is $(N,\mathbb{P})$-generic. So if $G$ is a $V$-generic filter on $\mathbb{P}$, then $N[G]\cap V=N$. In particular, $N[G]\cap\kappa=N\cap\kappa=\kappa_{N}\in S$. Let $\pi:N[G]\to\overline{N[G]}$ be the transitive collapsing map of $N[G]$ in $V[G]$. Let $G^{*}:=G\cap\mathbb{P}_{\kappa_{N}}$, which is a $V$-generic filter on $\mathbb{P}_{\kappa_{N}}$.

Lemma 2.2**.**

The following statements hold.

1. (1)

$\pi\upharpoonright N:N\to\overline{N}$* is the transitive collapsing map of $N$ in $V$;* 2. (2)

$\pi(\mathbb{P})=\mathbb{P}_{\kappa_{N}}$, $\pi(G)=G^{*}$, and $\overline{N[G]}=\overline{N}[G^{*}]$; in particular, $\overline{N[G]}$ is a member of $V[G^{*}]$; 3. (3)

$\overline{N[G]}=\overline{N}[G^{*}]$* is closed under $<\kappa_{N}$-sequences in $V[G^{*}]$.*

Proof.

(1) and (2) are straightforward. Since $\overline{N}^{<\kappa_{N}}\subseteq\overline{N}$ in $V$ by the closure of $N$ and $\mathbb{P}_{\kappa_{N}}$ is $\kappa_{N}$-c.c., (3) follows immediately by a standard fact. ∎

Now we are ready to state our second inductive hypothesis: for all $\beta<\alpha$ and for all $N\in\mathcal{X}_{\beta}$, letting $\pi:N[G]\to\overline{N[G]}$ be the transitive collapsing map of $N[G]$ and $G^{*}:=\pi(G)$, for all $q\in\pi(\mathbb{A}_{\beta})$, the forcing poset $\pi(\mathbb{A}_{\beta})/q$ is forcing equivalent to $\textrm{Add}(\omega_{2})$ in $V[G^{*}]$.

We begin the proof of the two inductive hypotheses for $\alpha$, assuming that they hold for all $\beta<\alpha$. Let $N\in\mathcal{X}_{\alpha}$. Let $\pi:N[G]\to\overline{N[G]}$ be the transitive collapsing map of $N[G]$ and $G^{*}:=\pi(G)$. Since $\pi$ is an isomorphism, by the absoluteness of suitable iterations we have that in $V[G^{*}]$,

[TABLE]

is a suitable iteration of length $\pi(\alpha)<\omega_{3}$. Applying Lemma 2.1 to this suitable iteration in the model $V[G^{*}]$, the second inductive hypothesis for $\alpha$ will follow from the next lemma.

Lemma 2.3**.**

For all $\gamma\in N\cap\alpha$, $\pi(\mathbb{A}_{\gamma})=\mathbb{A}_{\pi(\gamma)}^{*}$ forces over $V[G^{*}]$ that $\pi(\dot{T}_{\gamma})=\dot{T}_{\pi(\gamma)}^{*}$ is non-stationary in $\kappa_{N}$.

Proof.

Consider $\gamma\in N\cap\alpha$. Then by the choice of the names used in the iteration, $\mathbb{A}_{\gamma}$ forces that $\dot{T}_{\gamma}$ is a subset of $\kappa\cap\mathrm{cof}(\omega)$ which does not reflect to any ordinal in $\kappa\cap\mathrm{cof}(\omega_{1})$. In particular, $\mathbb{A}_{\gamma}$ forces that $\dot{T}_{\gamma}\cap\kappa_{N}$ is non-stationary in $\kappa_{N}$.

Consider $q\in\pi(\mathbb{A}_{\gamma})$. We will find a $V[G^{*}]$-generic filter $H$ on $\pi(\mathbb{A}_{\gamma})$ which contains $q$ such that in $V[G^{*}][H]$, $\pi(\dot{T}_{\gamma})^{H}$ is non-stationary in $\kappa_{N}$. Because $q$ is arbitrary, this proves that $\pi(\mathbb{A}_{\gamma})$ forces that $\pi(\dot{T}_{\gamma})$ is non-stationary. Since $N$ is in $\mathcal{X}_{\alpha}$ and $\gamma\in N\cap\alpha$, $N$ is in $\mathcal{X}_{\gamma}$. By the second inductive hypothesis, $\pi(\mathbb{A}_{\gamma})/q$ is forcing equivalent to $\textrm{Add}(\kappa_{N})$ in $V[G^{*}]$. By definition, the forcing iteration $\mathbb{P}$ forces with $\textrm{Add}(\kappa_{N})$ at stage $\kappa_{N}$. Hence, we can write $V[G\cap\mathbb{P}_{\kappa_{N}+1}]$ as $V[G^{*}][H]$, where $H$ is some $V[G^{*}]$-generic filter on $\pi(\mathbb{A}_{\gamma})/q$.

Now $\pi\upharpoonright\mathbb{A}_{\gamma}$ is an isomorphism between the posets $N[G]\cap\mathbb{A}_{\gamma}$ and $\pi(\mathbb{A}_{\gamma})$. Therefore, $I:=\pi^{-1}(H)$ is a filter on $N[G]\cap\mathbb{A}_{\gamma}$. The fact that $H$ is a $V[G^{*}]$-generic filter on $\pi(\mathbb{A}_{\gamma})$ easily implies that $I$ meets every dense subset of $\mathbb{A}_{\gamma}$ which is a member of $N[G]$. Now a lower bound $t$ of $I$ can be easily constructed by taking the coordinate-wise closure of the union of the clubs appearing in the conditions of $I$. Namely, the fact that $I$ meets every dense set in $N[G]$ implies that the maximum member of any such club is equal to $\kappa_{N}$, which has cofinality $\omega_{1}$ in $V[G]$ and hence is not in any of the sets $\dot{T}_{j}$.

Fix a $V[G]$-generic filter $h$ on $\mathbb{A}_{\gamma}$ which contains $t$. Now $\pi^{-1}:\overline{N}[G^{*}]\to N[G]$ is an elementary embedding of $\overline{N}[G^{*}]$ into $H(\kappa^{+})^{V[G]}$ which satisfies that $\pi^{-1}(H)=I\subseteq h$. So by a standard fact about extending elementary embeddings, we can extend $\pi^{-1}$ to an elementary embedding $\tau:\overline{N}[G^{*}][H]\to N[G][h]$ which maps $H$ to $h$. Let $T^{*}:=\pi(\dot{T}_{\gamma})^{H}$ and $T_{\gamma}:=(\dot{T}_{\gamma})^{h}$. Then clearly, $\tau(T^{*})=T_{\gamma}$.

Since $\kappa_{N}$ is the critical point of $\tau$, $T_{\gamma}\cap\kappa_{N}=T^{*}$. As $\mathbb{A}_{\gamma}$ forces that $\dot{T}_{\gamma}$ does not reflect to $\kappa_{N}$, $T^{*}$ is a non-stationary subset of $\kappa_{N}$ in the model $V[G][h]$. By the first inductive hypothesis, $\mathbb{A}_{\gamma}$ is $\kappa$-distributive. Therefore, any club of $\kappa_{N}$ in $V[G][h]$ is actually in $V[G]$. Thus, $T^{*}$ is non-stationary in $V[G]$. But $V[G]$ is a generic extension of $V[G^{*}][H]$ by the proper forcing $\mathbb{P}_{\kappa_{N}+1,\kappa}$. So $T^{*}$ is non-stationary in $V[G^{*}][H]$. ∎

This completes the proof of the second inductive hypothesis. It remains to prove the first inductive hypothesis that $\mathbb{A}_{\alpha}$ is $\kappa$-distributive and preserves the stationarity of $S$.

Lemma 2.4**.**

For all $N\in\mathcal{X}_{\alpha}$, for all $a\in N[G]\cap\mathbb{A}_{\alpha}$, there exists a filter $I$ on $N[G]\cap\mathbb{A}_{\alpha}$ in $V[G]$ containing $a$ which meets every dense subset of $\mathbb{A}_{\alpha}$ in $N[G]$.

Proof.

This is similar to a part of the proof of the Lemma 2.3. Let $\pi:N[G]\to\overline{N[G]}$ be the transitive collapsing map of $N[G]$ and $G^{*}:=\pi(G)$. Let $a\in N[G]\cap\mathbb{A}_{\alpha}$. Then $\pi(a)\in\pi(\mathbb{A}_{\alpha})$. By the second inductive hypothesis which we have now verified for $\alpha$, $\pi(\mathbb{A}_{\alpha})/\pi(a)$ is forcing equivalent to $\textrm{Add}(\kappa_{N})$ in $V[G^{*}]$. By definition, the forcing iteration $\mathbb{P}$ forces with $\textrm{Add}(\kappa_{N})$ at stage $\kappa_{N}$. Hence, we can write $V[G\cap\mathbb{P}_{\kappa_{N}+1}]$ as $V[G^{*}][H]$, where $H$ is some $V[G^{*}]$-generic filter on $\pi(\mathbb{A}_{\alpha})/\pi(a)$.

Now $\pi\upharpoonright\mathbb{A}_{\alpha}$ is an isomorphism between the posets $N[G]\cap\mathbb{A}_{\alpha}$ and $\pi(\mathbb{A}_{\alpha})$. Therefore, $I:=\pi^{-1}(H)$ is a filter on $N[G]\cap\mathbb{A}_{\alpha}$. The fact that $H$ is a $V[G^{*}]$-generic filter on $\pi(\mathbb{A}_{\alpha})$ easily implies that $I$ meets every dense subset of $\mathbb{A}_{\alpha}$ which is a member of $N[G]$. ∎

We can now complete the proof that $\mathbb{A}_{\alpha}$ is $\kappa$-distributive and preserves the stationarity of $S$. Given a family $\mathcal{D}$ of fewer than $\kappa$ many dense open subsets of $\mathbb{A}_{\alpha}$ and a condition $a\in\mathbb{A}_{\alpha}$, we may pick $N\in\mathcal{X}_{\alpha}$ so that $\mathcal{D}$ and $a$ are members of $N[G]$. Then $\mathcal{D}\subseteq N[G]$. By Lemma 2.4, fix a filter $I$ on $N[G]\cap\mathbb{A}_{\alpha}$ in $V[G]$ which contains $a$ and meets every dense subset of $\mathbb{A}_{\alpha}$ in $N[G]$ (and in particular, meets every dense set in $\mathcal{D}$). It is easy to define a lower bound $t$ of $I$ in $\mathbb{A}_{\alpha}$ by taking the coordinate-wise closure of the union of the clubs appearing in the conditions in $I$. Then $t\leq a$ and $t$ is in every dense open set in $\mathcal{D}$.

Similarly, given an $\mathbb{A}_{\alpha}$-name $\dot{C}$ for a club subset of $\kappa$ and $a\in\mathbb{A}_{\alpha}$, we may choose $N\in\mathcal{X}_{\alpha}$ such that $\dot{C}$ and $a$ are in $N$. Fix a filter $I$ on $N[G]\cap\mathbb{A}_{\alpha}$ in $V[G]$ which contains $a$ and meets every dense subset of $\mathbb{A}_{\alpha}$ in $N[G]$. As usual, let $t$ be a lower bound of $I$. Then $t$ is an $(N[G],\mathbb{A}_{\alpha})$-generic condition, which implies that $t$ forces that $N[G]\cap\kappa=\kappa_{N}$ is in $S\cap\dot{C}$.

3. Arbitrarily large continuum

In the model of the previous section, $2^{\omega}=\omega_{2}$ holds. A violation of CH is necessary, since CH implies the $\omega_{1}$-approximation property, as witnessed by any enumeration of all countable subsets of $\omega_{2}$ in order type $\omega_{2}$. In this section, we will show how to modify this model to obtain arbitrarily large continuum. This modification will use an unpublished result of I. Neeman.

Theorem 3.1** (Neeman).**

Assume that stationary reflection holds at $\omega_{2}$. Then for any ordinal $\mu$, $\mathrm{Add}(\omega,\mu)$ forces that stationary reflection still holds at $\omega_{2}$.

Proof.

We first prove the result in the special case that $\mu=\omega_{2}$. Let $p\in\textrm{Add}(\omega,\omega_{2})$, and suppose that $p$ forces that $\dot{S}$ is a stationary subset of $\omega_{2}\cap\mathrm{cof}(\omega)$. We will find $q\leq p$ and an ordinal $\beta\in\omega_{2}\cap\mathrm{cof}(\omega_{1})$ such that $q$ forces that $\dot{S}\cap\beta$ is stationary in $\beta$.

Let $T$ be the set of ordinals $\alpha<\omega_{2}$ such that for some $s\leq p$, $s$ forces that $\alpha\in\dot{S}$. Then $T\subseteq\omega_{2}\cap\mathrm{cof}(\omega)$. An easy observation is that $p$ forces that $\dot{S}\subseteq T$, and consequently $T$ is a stationary subset of $\omega_{2}$. For each $\alpha\in T$, fix a witness $s_{\alpha}\leq p$ which forces that $\alpha\in\dot{S}$, and define

[TABLE]

Using Fodor’s lemma, we can find a stationary set $U\subseteq T$ and a set $x$ satisfying that for all $\alpha\in U$, $a_{\alpha}=x$. Observe that $q:=x\cup p$ is a condition which extends $p$. Applying the fact that stationary reflection holds in the ground model together with an easy closure argument, we can fix $\beta\in\omega_{2}\cap\mathrm{cof}(\omega_{1})$ such that $U\cap\beta$ is stationary in $\beta$ and for all $\alpha<\beta$, $\mathrm{dom}(s_{\alpha})\subseteq\beta\times\omega$.

We claim that $q$ forces that $\dot{S}\cap\beta$ is stationary in $\beta$, which finishes the proof. Suppose for a contradiction that there is $r\leq q$ which forces that $\dot{S}\cap\beta$ is non-stationary in $\beta$. Using the fact that $\textrm{Add}(\omega,\omega_{2})$ is c.c.c. and $\mathrm{cf}(\beta)=\omega_{1}$, there exists a club $D\subseteq\beta$ in the ground model such that $r$ forces that $D\cap\dot{S}=\emptyset$. As $r$ is finite, we can fix $\delta<\beta$ such that $\mathrm{dom}(r)\cap(\beta\times\omega)\subseteq\delta\times\omega$.

Since $U\cap\beta$ is stationary in $\beta$, fix $\alpha\in U\cap D$ larger than $\delta$. We claim that $s_{\alpha}$ and $r$ are compatible. By the choice of $U$, $s_{\alpha}\upharpoonright(\alpha\times\omega)=x$, and by the choice of $\beta$, $\mathrm{dom}(s_{\alpha})\subseteq\beta\times\omega$. Suppose that $(\xi,n)\in\mathrm{dom}(s_{\alpha})\cap\mathrm{dom}(r)$. Then $\xi<\beta$, so $(\xi,n)\in\mathrm{dom}(r)\cap(\beta\times\omega)\subseteq\delta\times\omega$. Thus, $\xi<\delta<\alpha$. So $(\xi,n)\in\alpha\times\omega$, and hence $s_{\alpha}(\xi,n)=a_{\alpha}(\xi,n)=x(\xi,n)$. On the other hand, $r\leq q\leq x$, and so $r(\xi,n)=x(\xi,n)=s_{\alpha}(\xi,n)$.

This proves that $r$ and $s_{\alpha}$ are compatible. Fix $t\leq r,s_{\alpha}$. Since $t\leq s_{\alpha}$, $t$ forces that $\alpha\in\dot{S}$. On the other hand, $\alpha\in D$, and $r$ forces that $\dot{S}\cap D=\emptyset$. So $r$, and hence $t$, forces that $\alpha\notin\dot{S}$, which is a contradiction.

Now we prove the result for arbitrary ordinals $\mu$. If $\mu<\omega_{2}$, then $\textrm{Add}(\omega,\omega_{2})$ is isomorphic to $\textrm{Add}(\omega,\mu)\times\textrm{Add}(\omega,\omega_{2}\setminus\mu)$. Since stationary reflection holds in $V^{\textrm{Add}(\omega,\omega_{2})}$, it also holds in the submodel $V^{\textrm{Add}(\omega,\mu)}$, since a non-reflecting stationary set in the latter model would remain a non-reflecting stationary set in the former model.

Suppose that $\mu>\omega_{2}$. Let $p$ be a condition in $\textrm{Add}(\omega,\mu)$ which forces that $\dot{S}$ is a stationary subset of $\omega_{2}\cap\mathrm{cof}(\omega)$, for some nice name $\dot{S}$. Then by the c.c.c. property of $\textrm{Add}(\omega,\mu)$ and the fact that conditions are finite, it is easy to show there exists a set $X\subseteq\mu$ of size $\omega_{2}$ such that $\dot{S}$ is a nice $\textrm{Add}(\omega,X)$-name and $p\in\textrm{Add}(\omega,X)$. Since $X$ has size $\omega_{2}$, $\textrm{Add}(\omega,X)$ is isomorphic to $\textrm{Add}(\omega,\omega_{2})$. By the first result above, we can find $q\leq p$ in $\textrm{Add}(\omega,X)$ and $\beta\in\omega_{2}\cap\mathrm{cof}(\omega_{1})$ such that $q$ forces in $\textrm{Add}(\omega,X)$ that $\dot{S}\cap\beta$ is stationary in $\beta$. Since $\textrm{Add}(\omega,\mu)$ is isomorphic to $\textrm{Add}(\omega,X)\times\textrm{Add}(\omega,\mu\setminus X)$ and $\textrm{Add}(\omega,\mu\setminus X)$ is c.c.c. in $V^{\textrm{Add}(\omega,X)}$, an easy argument shows that $q$ forces in $\textrm{Add}(\omega,\mu)$ that $\dot{S}\cap\beta$ is stationary in $\beta$. ∎

Now start with the model $W:=V^{\mathbb{P}*\dot{\mathbb{A}}}$ from the previous section. Then $\omega_{2}$ is not weakly compact in $L$, there exists a disjoint stationary sequence in $W$, and stationary reflection holds at $\omega_{2}$ in $W$. Let $\mu$ be any ordinal and let $H$ be a $W$-generic filter on $\textrm{Add}(\omega,\mu)$. Since $\textrm{Add}(\omega,\mu)$ is c.c.c., Corollary 1.3 implies that there exists a disjoint stationary sequence in $W[H]$. As $\omega_{2}$ is not weakly compact in $L$, there exists an $\omega_{2}$-Aronszajn tree in $W[H]$. And stationary reflection holds in $W[H]$ by Theorem 3.1.