Martin's Maximum and the Diagonal Reflection Principle

TL;DR

This paper demonstrates that Martin's Maximum, a strong forcing axiom, does not imply the Diagonal Reflection Principle for stationary subsets of  , clarifying the independence of these set-theoretic principles.

Contribution

The paper establishes the independence of the Diagonal Reflection Principle from Martin's Maximum, showing they are not logically equivalent.

Findings

Martin's Maximum does not imply the Diagonal Reflection Principle.

The result clarifies the relationship between forcing axioms and reflection principles.

Provides new insights into the hierarchy of set-theoretic axioms.

Abstract

We prove that Martin's Maximum does not imply the Diagonal Reflection Principle for stationary subsets of $[ \omega_2 ]^\omega$.

Equations

$$Z:=\{x\in[\delta^{*}]^{\omega}\mid\mbox{$x\subseteq\delta_{\xi}=\sup(x)$ for some limit $\xi<\omega_{1}$}\}$$

$$\{x\in [{\omega }_2{]}^{\omega }\mid \exists p\in \dot{G},p(\alpha ,x)=1\}.$$

$$\mathbb{S}:=\mathbb{P}\ast \dot{\mathbb{Q}}\ast \mathbb{R}([{\omega }_2^V{]}^{\omega }\setminus {\dot{X}}_{\dot{\gamma }}),$$

$$g:=\{p\in \mathbb{P}\cap M\mid \exists \dot{q}\exists \dot{r},p\ast \dot{q}\ast \dot{r}\in k\}.$$

Full text

1 Introduction

In Foreman-Magidor-Shelah [5], it was shown that Martin’s Maximum $\mathsf{MM}$ implies the following stationary reflection principle, which is called the Weak Reflection Principle:

• $\mathsf{WRP}\equiv$

For any cardinal $\lambda\geq\omega_{2}$ and any stationary $X\subseteq[\lambda]^{\omega}$, there is $R\in[\lambda]^{\omega_{1}}$ with $R\supseteq\omega_{1}$ such that $X\cap[R]^{\omega}$ is stationary in $[R]^{\omega}$.

$\mathsf{WRP}$ is known to have many interesting cosequences such as Chang’s Conjecture (Foreman-Magidor-Shelah [5]), the presaturation of the non-stationary ideal over $\omega_{1}$ (Feng-Magidor [4]), $2^{\omega}\leq\omega_{2}$ (folklore) and the Singular Cardinal Hypothesis (Shelah [12]).

As for stationary reflection principles, simultaneous reflection is often discussed. Larson [10] proved that $\mathsf{MM}$ also implies the following simultaneous reflection principle of $\omega_{1}$-many stationary sets:

• $\mathsf{WRP}_{\omega_{1}}\equiv$

For any cardinal $\lambda\geq\omega_{2}$ and any sequence $\langle X_{\xi}\mid\xi<\omega_{1}\rangle$ of stationary subsets of $[\lambda]^{\omega}$, there is $R\in[\lambda]^{\omega_{1}}$ with $R\supseteq\omega_{1}$ such that $X_{\xi}\cap[R]^{\omega}$ is stationary in $[R]^{\omega}$ for all $\xi<\omega_{1}$.

Cox [2] formulated the following strengthening of $\mathsf{WRP}_{\omega_{1}}$, which is called the Diagonal Reflection Principle:

• $\mathsf{DRP}\equiv$

For any cardinal $\lambda\geq\omega_{2}$ and any sequence $\langle X_{\alpha}\mid\alpha<\lambda\rangle$ of stationary subsets of $[\lambda]^{\omega}$, there is $R\in[\lambda]^{\omega_{1}}$ with $R\supseteq\omega_{1}$ such that $X_{\alpha}\cap[R]^{\omega}$ is stationary in $[R]^{\omega}$ for all $\alpha\in R$.

Recently, Fuchino-Ottenbreit-Sakai [6] proved that a variation of $\mathsf{DRP}$ is equivalent to some variation of the downward Löwenheim-Skolem theorem of the stationary logic. Cox [2] also introduced the following weakning of $\mathsf{DRP}$, where $X\subseteq[\lambda]^{\omega}$ is said to be projectively stationary if the set $\{x\in X\mid x\cap\omega_{1}\in S\}$ is stationary in $[\lambda]^{\omega}$ for any stationary $S\subseteq\omega_{1}$:

• $\mathsf{wDRP}\equiv$

For any cardinal $\lambda\geq\omega_{2}$ and any sequence $\langle X_{\alpha}\mid\alpha<\lambda\rangle$ of projectively stationary subsets of $[\lambda]^{\omega}$, there is $R\subseteq[\lambda]^{\omega_{1}}$ with $R\supseteq\omega_{1}$ such that $X_{\alpha}\cap[R]^{\omega}$ is stationary in $[R]^{\omega}$ for all $\alpha\in R$.

Cox [2] proved that $\mathsf{MM}$ implies $\mathsf{wDRP}$, but it remained open whether $\mathsf{MM}$ implies $\mathsf{DRP}$. In this paper, we prove that $\mathsf{MM}$ does not imply $\mathsf{DRP}$. In fact, we prove slightly more.

To state our result, we recall $+$-versions of the forcing axiom. For a class $\Gamma$ of forcing notions and a cardinal $\mu\leq\omega_{1}$, $\mathsf{MA}^{+\mu}(\Gamma)$ is the following statement:

• $\mathsf{MA}^{+\mu}(\Gamma)\equiv$

For any $\mathbb{P}\in\Gamma$, any sequence $\langle D_{\xi}\mid\xi<\omega_{1}\rangle$ of dense subsets of $\mathbb{P}$ and any sequence $\langle\dot{S}_{\eta}\mid\eta<\mu\rangle$ of $\mathbb{P}$-names of stationary subsets of $\omega_{1}$, there is a filter $g\subseteq\mathbb{P}$ such that

1. (i)

$g\cap D_{\xi}\neq\emptyset$ for any $\xi<\omega_{1}$, 2. (ii)

$\dot{S}_{\eta}^{g}=\{\alpha<\omega_{1}\mid\exists p\in g,\ p\Vdash_{\mathbb{P}}\textrm{``}\alpha\in\dot{S}_{\eta}\textrm{''}\}$ is stationary in $\omega_{1}$ for all $\eta<\mu$.

Let $\mathsf{MA}^{+\mu}(\sigma\mbox{-closed})$ denote $\mathsf{MA}^{+\mu}(\Gamma)$ for the class $\Gamma$ of all $\sigma$-closed forcing notions. Also, let $\mathsf{MM}^{+\mu}$ denote $\mathsf{MA}^{+\mu}(\Gamma)$ for the class $\Gamma$ of all $\omega_{1}$-stationary preserving forcing notions. It is well-known that $\mathsf{MA}^{+\omega_{1}}(\sigma\mbox{-closed})$ holds if a supercompact cardinal is Lévy collapsed to $\omega_{2}$ and that $\mathsf{MM}^{+\omega_{1}}$ holds in the standard model of $\mathsf{MM}$ constructed in Foreman-Magidor-Shelah [5].

Cox [2] proved that $\mathsf{MA}^{+\omega_{1}}(\sigma\mbox{-closed})$ implies $\mathsf{DRP}$. So $\mathsf{MM}^{+\omega_{1}}$ implies $\mathsf{DRP}$. In this paper, we prove that $\mathsf{MM}^{+\omega}$ does not imply $\mathsf{DRP}$:

Main Theorem**.**

Assume $\mathsf{MM}^{+\omega}$ holds. Then there is a forcing extension in which $\mathsf{MM}^{+\omega}$ remains to hold, but $\mathsf{DRP}$ fails at $[\omega_{2}]^{\omega}$.

Our proof of the Main Theorem is based on the proof of the classical result, due to Beaudoin [1] and Magidor, that the Proper Forcing Axiom does not imply the reflection of stationary subsets of the set $\{\alpha\in\omega_{2}\mid\mathrm{cof}(\alpha)=\omega\}$. Similar arguments are used in König-Yoshinobu [8], Yoshinobu [13], [14] and Cox [3], to separate reflection principles from strong forcing axioms.

We will prove the Main Theorem in Section 3. In Section 2, we will present our notation and basic facts used in this paper.

2 Preliminaries

Here we present our notation and basic facts. See Jech [7] for those which are not mentioned here.

First, we recall the notion of stationary sets in $[W]^{\omega}$. Let $W$ be a set with $\omega_{1}\subseteq W$. $Z\subseteq[W]^{\omega}$ is said to be club in $[W]^{\omega}$ if $Z$ is $\subseteq$-cofinal in $[W]^{\omega}$, and $\bigcup_{n\in\omega}x_{n}\in Z$ for any $\subseteq$-increasing sequence $\langle x_{n}\mid n<\omega\rangle$ of elements of $Z$. $X\subseteq[W]^{\omega}$ is said to be stationary in $[W]^{\omega}$ if $X\cap Z\neq\emptyset$ for any club $Z\subseteq[W]^{\omega}$. For $S\subseteq\omega_{1}$, $S$ is stationary in $\omega_{1}$ in the usual sense if and only if $S$ is stationary in $[\omega_{1}]^{\omega}$ in the above sense.

We will use the following standard facts without any reference. Proofs can be found also in Jech [7].

Fact 2.1** ((1) Kueker [9], (2) Menas [11]).**

Suppose $W$ is a set $\supseteq\omega_{1}$ and $X$ is a subset of $[W]^{\omega}$.

1. (1)

$X$* is stationary if and only if for any function $F:[W]^{<\omega}\to W$ there is a non-empty $x\in X$ which is closed under $F$, i.e. $F(a)\in x$ for all $a\in[x]^{<\omega}$.* 2. (2)

Suppose $W^{\prime}\supseteq W$. Then $X$ is stationary in $[W]^{\omega}$ if and only if the set $\{x^{\prime}\in[W^{\prime}]^{\omega}\mid x^{\prime}\cap W\in X\}$ is stationary in $[W^{\prime}]^{\omega}$.

Here we slightly simplify $\mathsf{DRP}$ at $[\omega_{2}]^{\omega}$.

Lemma 2.2**.**

Assume $\mathsf{DRP}$ at $[\omega_{2}]^{\omega}$. Then, for any sequence $\langle X_{\alpha}\mid\alpha<\omega_{2}\rangle$ of stationary subsets of $[\omega_{2}]^{\omega}$, there is $\delta\in\omega_{2}\setminus\omega_{1}$ such that $X_{\alpha}\cap[\delta]^{\omega}$ is stationary in $[\delta]^{\omega}$ for all $\alpha<\delta$.

Proof.

Suppose $\langle X_{\alpha}\mid\alpha<\omega_{2}\rangle$ is a sequence of stationary subsets of $[\omega_{2}]^{\omega}$. We find $\delta$ as in the lemma.

For each $\beta<\omega_{2}$, take a surjection $\pi_{\beta}:\omega_{1}\to\beta$. Let $Z$ be the set of all $x\in[\omega_{2}]^{\omega}$ such that $x\cap\omega_{1}\in\omega_{1}$ and $x$ is closed under $\pi_{\beta}$ for all $\beta\in x$. Then, $Z$ is club in $[\omega_{2}]^{\omega}$. Moreover, it is easy to see that if $\omega_{1}\subseteq R\in[\omega_{2}]^{\omega_{1}}$, and $Z\cap[R]^{\omega}$ is $\subseteq$-cofinal in $[R]^{\omega}$, then $R\in\omega_{2}\setminus\omega_{1}$.

By shrinking $X_{0}$ if necessary, we may assume that $X_{0}\subseteq Z$. By $\mathsf{DRP}$ at $[\omega_{2}]^{\omega}$, take $R\in[\omega_{2}]^{\omega_{1}}$ including $\omega_{1}$ such that $X_{\alpha}\cap[R]^{\omega}$ is stationary for all $\alpha\in R$. Then, $R\in\omega_{2}\setminus\omega_{1}$ since $Z\cap[R]^{\omega}$ is $\subseteq$-cofinal in $[R]^{\omega}$. So, $\delta:=R$ is as desired. ∎

Next, we present our notation and basic facts about forcing. Suppose $\mathbb{P}$ is a forcing notion and $M$ is a set. We say that $g\subseteq\mathbb{P}\cap M$ is $M$-generic if $g\cap D\neq\emptyset$ for any dense $D\subseteq\mathbb{P}$ with $D\in M$.

We will use the following well-known fact about forcing axioms:

Fact 2.3** (Woodin [15]).**

Let $\Gamma$ be a class of forcing notions and $\mu$ be a cardinal $\leq\omega_{1}$, and assume $\mathsf{MA}^{+\mu}(\Gamma)$ holds. Suppose $\mathbb{P}\in\Gamma$ and $\langle\dot{T}_{\xi}\mid\xi<\mu\rangle$ is a sequence of $\mathbb{P}$-names for stationary subsets of $\omega_{1}$. Then, for any regular cardinal $\theta$ with $\mathbb{P}\in\mathcal{H}_{\theta}$ and any $A\in[\mathcal{H}_{\theta}]^{\omega_{1}}$, there are $M\in[\mathcal{H}_{\theta}]^{\omega_{1}}$ and $g\subseteq\mathbb{P}\cap M$ with the following properties.

1. (i)

$A\subseteq M\prec\langle\mathcal{H}_{\theta},\in\rangle$. 2. (ii)

$g$* is an $M$-generic filter on $\mathbb{P}\cap M$.* 3. (iii)

$\dot{T}_{\xi}^{g}$* is stationary in $\omega_{1}$ for any $\xi<\mu$.*

We will also use forcing notions for shooting club sets. For an ordinal $\lambda\geq\omega_{1}$ and a subset $X$ of $[\lambda]^{\omega}$, let $\mathbb{R}(X)$ denote the poset of all $\subseteq$-increasing continuous function from some countable successor ordinal to $X$, which is ordered by reverse inclusions. The following is standard:

Lemma 2.4**.**

Suppose $X$ is a stationary subset of $[\lambda]^{\omega}$ for some ordinal $\lambda\geq\omega_{1}$.

1. (1)

A forcing extension by $\mathbb{R}(X)$ adds no new countable sequences of ordinals. So it preserves $\omega_{1}$. 2. (2)

In $V^{\mathbb{R}(X)}$, $X$ contains a club subset of $[\lambda]^{\omega}$. 3. (3)

In $V$, suppose $Y\subseteq X$ and $Y$ is stationary in $[\lambda]^{\omega}$. Then $Y$ remains stationary in $V^{\mathbb{R}(X)}$.

Proof.

Let $\mathbb{R}$ denote $\mathbb{R}(X)$. Before starting, note that the set $\{r\in\mathbb{R}\mid\exists\xi\in\mathop{\mathrm{dom}}\nolimits(r),\ r(\xi)\supseteq x\}$ is dense in $\mathbb{R}$ for any $x\in[\lambda]^{\omega}$, since $X$ is $\subseteq$-cofinal in $[\lambda]^{\omega}$.

First, we prove (1) and (3). We work in $V$. Suppose $r\in\mathbb{R}$, $\mathcal{D}$ is a countable family of dense open subsets of $\mathbb{R}$ and $\dot{F}$ is an $\mathbb{R}$-name for a function from $[\lambda]^{<\omega}$ to $\lambda$. It suffices to find $r^{*}\leq r$ and $y\in Y$ such that $r^{*}\in\bigcap\mathcal{D}$ and $r^{*}$ forces $y$ to be closed under $\dot{F}$.

Take a sufficiently large regular cardinal $\theta$. Since $Y$ is stationary, there is a countable $M\prec\langle\mathcal{H}_{\theta},\in\rangle$ such that $\{\lambda,X,r,\dot{F}\}\cup\mathcal{D}\subseteq M$ and $y:=M\cap\lambda\in Y$. Then, we can construct a descending sequence $\langle r_{n}\mid n<\omega\rangle$ in $\mathbb{R}\cap M$ such that $r_{0}=r$ and $\{r_{n}\mid n<\omega\}$ is $M$-generic. Note that any lower bound of $\{r_{n}\mid n<\omega\}$ forces $y$ to be closed under $\dot{F}$ by the $M$-genericity of $\{r_{n}\mid n<\omega\}$.

Let $r^{\prime}:=\bigcup_{n<\omega}r_{n}$ and $\zeta:=\mathop{\mathrm{dom}}\nolimits(r^{\prime})$. Then, using the fact mentioned at the beginning, it is easy to check that $\zeta$ is a limit ordinal and $\bigcup_{\xi<\zeta}r^{\prime}(\xi)=y$. Let $r^{*}$ be an extension of $r^{\prime}$ such that $\mathop{\mathrm{dom}}\nolimits(r^{*})=\zeta+1$ and $r^{*}(\zeta)=y$. Then $r^{*}\in\mathbb{R}$, and $r^{*}$ is a lower bound of $\{r_{n}\mid n<\omega\}$. So $r^{*}$ and $y$ are as desired.

Next, we check (2). By (1), the definition of $\mathbb{R}$ and the fact mentioned at the beginning, if $G$ is an $\mathbb{R}$-generic filter over $V$, then $\mathrm{range}(\bigcup G)$ is a club subset of $[\lambda]^{\omega}$ consisting of elements of $X$. So (2) holds. ∎

3 Proof of Main Theorem

Here we prove the Main Theorem. Throughout this section, assume that $\mathsf{MM}^{+\omega}$ holds in the ground model $V$.

We construct a forcing notion which preserves $\mathsf{MM}^{+\omega}$ and adds a counter-example $\langle X_{\alpha}\mid\alpha<\omega_{2}\rangle$ of the consequence of Lemma 2.2. Here recall that $\mathsf{MM}$ implies $\mathsf{wDRP}$. So we must arrange our forcing notion so that each $X_{\alpha}$ is not projectively stationary. For some technical reason, we also make $\langle X_{\alpha}\mid\alpha<\omega_{2}\rangle$ pairwise disjoint.

Recall the fact, due to Foreman-Magidor-Shelah [5], that $\mathsf{MM}$ implies $2^{\omega_{1}}=\omega_{2}$. In $V$, fix an enumeration $\langle S_{\alpha}\mid\alpha<\omega_{2}\rangle$ of all stationary subsets of $\omega_{1}$. Let $\mathbb{P}$ be the following forcing notion:

• •

$\mathbb{P}$ consists of all functions $p$ such that

1. (i)

$p:\delta_{p}\times[\delta_{p}]^{\omega}\to 2$ for some $\delta_{p}<\omega_{2}$, 2. (ii)

for any $\alpha<\delta_{p}$, $X_{p,\alpha}:=\{x\in[\delta_{p}]^{\omega}\mid p(\alpha,x)=1\}$ has size $\leq\omega_{1}$, 3. (iii)

$x\cap\omega_{1}\in S_{\alpha}$ for any $\alpha<\delta_{p}$ and any $x\in X_{p,\alpha}$, 4. (iv)

$X_{p,\alpha}\cap X_{p,\beta}=\emptyset$ for any distinct $\alpha,\beta<\delta_{p}$, 5. (v)

for any $\delta\in\delta_{p}+1\setminus\omega_{1}$, there is $\alpha<\delta$ with $X_{p,\alpha}\cap[\delta]^{\omega}$ non-stationary in $[\delta]^{\omega}$.

• •

$p\leq p^{\prime}$ in $\mathbb{P}$ if $p\supseteq p^{\prime}$.

We observe basic properties of $\mathbb{P}$. Note that a forcing extension by $\mathbb{P}$ preserves all cardinals by (1) and (3) of the following lemma.

Lemma 3.1**.**

1. (1)

$|\mathbb{P}|=\omega_{2}$. 2. (2)

$\mathbb{P}$* is $\sigma$-closed.* 3. (3)

A forcing extension by $\mathbb{P}$ adds no new sequences of ordinals of length $\omega_{1}$. 4. (4)

For any $p\in\mathbb{P}$ and any $\delta<\omega_{2}$, there is $p^{\prime}\leq p$ with $\delta\leq\delta_{p^{\prime}}$.

Proof.

(1) This is clear from the definition of $\mathbb{P}$, especially the property (ii) of its conditions, and the fact that $2^{\omega_{1}}=\omega_{2}$ in $V$.

(4) Suppose $p\in\mathbb{P}$ and $\delta<\omega_{2}$. We may assume $\delta_{p}\leq\delta$. Let $p^{\prime}:\delta\times[\delta]^{\omega}\to 2$ be an extension of $p^{\prime}$ such that $p^{\prime}(\alpha,x)=0$ for all $\langle\alpha,x\rangle\notin\delta_{p}\times[\delta_{p}]^{\omega}$. It suffices to prove that $p^{\prime}\in\mathbb{P}$. We only check that $p^{\prime}$ satisfies the property (v) of conditions of $\mathbb{P}$. The other properties are easily checked.

Take an arbitrary $\gamma\in\delta+1\setminus\omega_{1}$. We find $\alpha<\delta$ with $X_{p^{\prime},\alpha}\cap[\gamma]^{\omega}$ is non-stationary. If $\gamma\leq\delta_{p}$, then we can find such $\alpha$ since $p\in\mathbb{P}$ and $p\subseteq p^{\prime}$. Suppose $\gamma>\delta_{p}$. Then $Z:=[\gamma]^{\omega}\setminus[\delta_{p}]^{\omega}$ is club in $[\gamma]^{\omega}$, and $X_{p^{\prime},\alpha}\cap Z=\emptyset$ for any $\alpha<\gamma$. So any $\alpha<\gamma$ is as desired in this case.

(2) Suppose $\langle p_{n}\mid n<\omega\rangle$ is a descending sequence in $\mathbb{P}$. We find a lower bound $p^{*}$ of $\{p_{n}\mid n<\omega\}$ in $\mathbb{P}$. We may assume that $\langle p_{n}\mid n<\omega\rangle$ is not eventually constant.

Let $\delta_{n}:=\delta_{p_{n}}$ for each $n<\omega$. Let $\delta^{*}:=\bigcup_{n<\omega}\delta_{n}$, and let $p^{*}:\delta^{*}\times[\delta^{*}]^{\omega}\to 2$ be an extension of $\bigcup_{n\in\omega}p_{n}$ such that $p^{*}(\alpha,x)=0$ for all $\alpha<\delta^{*}$ and all $x\in[\delta^{*}]^{\omega}\setminus\bigcup_{n\in\omega}[\delta_{n}]^{\omega}$. Note that $X_{p^{*},\alpha}$ is non-stationary in $[\delta^{*}]^{\omega}$ for any $\alpha<\delta^{*}$ since $Z:=[\delta^{*}]^{\omega}\setminus\bigcup_{n<\omega}[\delta_{n}]^{\omega}$ is club in $[\delta^{*}]^{\omega}$ and $X_{p^{*},\alpha}\cap Z=\emptyset$. Then it is easy to see that $p^{*}$ is as desired.

(3) Suppose $p\in\mathbb{P}$ and $\langle D_{\xi}\mid\xi<\omega_{1}\rangle$ is a sequence of dense open subsets of $\mathbb{P}$. It suffices to find $p^{*}\leq p$ with $p^{*}\in\bigcap_{\xi<\omega_{1}}D_{\xi}$.

We recursively construct a strictly descending sequence $\langle p_{\xi}\mid\xi<\omega_{1}\rangle$ in $\mathbb{P}$ as follows. For each $\xi<\omega_{1}$, we let $\delta_{\xi}$ denote $\delta_{p_{\xi}}$. First, let $p_{0}:=p$. If $p_{\xi}$ has been taken, then take $p_{\xi+1}<p_{\xi}$ with $p_{\xi+1}\in D_{\xi}$. Suppose $\xi$ is a limit ordinal $<\omega_{1}$ and $\langle p_{\eta}\mid\eta<\xi\rangle$ has been constructed. Then define $p_{\xi}$ as in the proof of (2). That is, let $\delta_{\xi}:=\bigcup_{\eta<\xi}\delta_{\eta}$, and let $p_{\xi}:\delta_{\xi}\times[\delta_{\xi}]^{\omega}\to 2$ be an extension of $\bigcup_{\eta<\xi}p_{\eta}$ such that $p_{\xi}(\alpha,x)=0$ for all $\alpha<\delta_{\xi}$ and all $x\in[\delta_{\xi}]^{\omega}\setminus\bigcup_{\eta<\xi}[\delta_{\eta}]^{\omega}$. Then $p_{\xi}$ is a lower bound of $\{p_{\eta}\mid\eta<\xi\}$ in $\mathbb{P}$.

We have constructed $\langle p_{\xi}\mid\xi<\omega_{1}\rangle$. Let $\delta^{*}:=\sup_{\xi<\omega_{1}}\delta_{\xi}$ and $p^{*}:=\bigcup_{\xi<\omega_{1}}p_{\xi}$. Here note that $[\delta^{*}]^{\omega}=\bigcup_{\xi<\omega_{1}}[\delta_{\xi}]^{\omega}$. So $p^{*}:\delta^{*}\times[\delta^{*}]^{\omega}\to 2$. Note also that $X_{p^{*},0}$ is non-stationary in $[\delta^{*}]^{\omega}$ since

[TABLE]

is club in $[\delta^{*}]^{\omega}$ and that $X_{p^{*},0}\cap Z=\emptyset$ by the construction of $p_{\xi}$ for a limit $\xi<\omega_{1}$. Then, it is easy to check that $p^{*}$ is as desired. ∎

Let $\dot{G}$ be the canonical $\mathbb{P}$-name for a $\mathbb{P}$-generic filter. For $\alpha<\omega_{2}$, let $\dot{X}_{\alpha}$ be the $\mathbb{P}$-name for the set

[TABLE]

Lemma 3.2**.**

For each $\alpha<\omega_{2}$, $\dot{X}_{\alpha}$ is stationary in $[\omega_{2}]^{\omega}$ in $V^{\mathbb{P}}$.

Proof.

We work in $V$. Take an arbitrary $\alpha<\omega_{2}$. Suppose $p\in\mathbb{P}$ and $\dot{F}$ is a $\mathbb{P}$-name for a function from $[\omega_{2}]^{<\omega}$ to $\omega_{2}$. It suffices to find $p^{*}\leq p$ and $x\in[\omega_{2}]^{<\omega}$ such that p^{*}\Vdash_{\mathbb{P}}\textrm{``}\,x\in\dot{X}_{\alpha}\wedge\mbox{x$is closed under$\dot{F}}\,\textrm{''}.

Take a sufficiently large regular cardinal $\theta$ and a countable $M\prec\langle\mathcal{H}_{\theta},\in\rangle$ such that $\alpha,\mathbb{P},p,\dot{F}\in M$ and $M\cap\omega_{1}=\alpha$. Let $x:=M\cap\omega_{2}$. We can take a descending sequence $\langle p_{n}\mid n<\omega\rangle$ in $\mathbb{P}\cap M$ such that $p_{0}=p$ and $\{p_{n}\mid n<\omega\}$ is $M$-generic. Note that any lower bound of $\{p_{n}\mid n<\omega\}$ forces $x$ to be closed under $\dot{F}$ by the $M$-genericity. For each $n<\omega$, let $\delta_{n}:=\delta_{p_{n}}$. Note that $\delta_{n}\in M\cap\omega_{2}$ for each $n<\omega$ and that $\delta^{*}:=\sup_{n<\omega}\delta_{n}=\sup(M\cap\omega_{2})$ by Lemma 3.1 (4).

Let $p^{*}:\delta^{*}\times[\delta^{*}]^{\omega}\to 2$ be an extension of $\bigcup_{n<\omega}p_{n}$ such that $p^{*}(\alpha,x)=1$ and $p^{*}(\beta,y)=0$ for any $\beta<\delta^{*}$ and any $y\in[\delta^{*}]^{\omega}\setminus\bigcup_{n<\omega}[\delta_{n}]^{\omega}$ with $\langle\beta,y\rangle\neq\langle\alpha,x\rangle$. Then, it is easy to check that $p^{*}$ and $x$ are as desired. ∎

The following is immediate from Lemma 2.2, 3.1, 3.2 and the property (v) of conditions of $\mathbb{P}$:

Corollary 3.3**.**

$\mathsf{DRP}$* at $\omega_{2}$ fails in $V^{\mathbb{P}}$.*

We must show that $\mathbb{P}$ preserves $\mathsf{MM}^{+\omega}$. The following lemma is a key:

Lemma 3.4**.**

Let $\dot{\mathbb{Q}}$ be a $\mathbb{P}$-name for an $\omega_{1}$-stationary preserving forcing notion and $\langle\dot{T}_{n}\mid n<\omega\rangle$ be a sequence of $\mathbb{P}*\dot{\mathbb{Q}}$-names for stationary subsets of $\omega_{1}$. Then there is a $\mathbb{P}*\dot{\mathbb{Q}}$-name $\dot{\gamma}$ of an ordinal $<\omega_{2}^{V}$ such that if we let

[TABLE]

then all elements of $\{S_{\alpha}\mid\alpha<\omega_{2}^{V}\}\cup\{\dot{T}_{n}\mid n<\omega\}$ remain stationary in $V^{\mathbb{S}}$.

Proof.

Let $\lambda:=\omega_{2}^{V}$. Suppose $G*H$ is a $\mathbb{P}*\dot{\mathbb{Q}}$-generic filter over $V$. In $V[G*H]$, let $X_{\alpha}:=\dot{X}_{\alpha}^{G}$ for $\alpha<\lambda$ and $T_{n}:=\dot{T}_{n}^{G*H}$ for $n<\omega$. Moreover, let $\mathbb{R}_{\alpha}$ denote $\mathbb{R}([\lambda]^{\omega}\setminus X_{\alpha})$ for $\alpha<\lambda$. In $V[G*H]$, we find $\gamma<\lambda$ such that $\mathbb{R}_{\gamma}$ forces all elements of $\{S_{\alpha}\mid\alpha<\lambda\}\cup\{T_{n}\mid n<\omega\}$ stationary. Here note that all $S_{\alpha}$ and $T_{n}$ are stationary in $V[G*H]$ by the fact that $\mathbb{P}*\dot{\mathbb{Q}}$ is $\omega_{1}$-stationary preserving and the assumption on $\langle\dot{T}_{n}\mid n<\omega\rangle$.

We work in $V[G*H]$. For $S\subseteq\omega_{1}$, let $\bar{S}:=\{x\in[\lambda]^{\omega}\mid x\cap\omega_{1}\in S\}$. For $X,Y\subseteq[\lambda]^{\omega}$, we write $X\subseteq^{*}Y$ if $X\setminus Y$ is non-stationary in $[\lambda]^{\omega}$. By Lemma 2.4, for $S\subseteq\omega_{1}$ and $\alpha<\lambda$, $\mathbb{R}_{\alpha}$ does not force $S\subseteq\omega_{1}$ stationary if and only if $\bar{S}\subseteq^{*}X_{\alpha}$.

Since $\langle X_{\alpha}\mid\alpha<\lambda\rangle$ is pairwise disjoint, for each $n<\omega$ there is at most one $\alpha<\lambda$ with $\bar{T}_{n}\subseteq^{*}X_{\alpha}$. Since $|\lambda|\geq\omega_{1}$, we can take $\beta<\lambda$ such that $\bar{T}_{n}\not\subseteq^{*}X_{\beta}$ for any $n<\omega$. Then $\mathbb{R}_{\beta}$ forces $T_{n}$ stationary for all $n<\omega$. Thus, if $\mathbb{R}_{\beta}$ also forces $S_{\alpha}$ stationary for all $\alpha<\lambda$, then $\gamma:=\beta$ is as desired.

Assume there is $\alpha<\lambda$ such that $\mathbb{R}_{\beta}$ does not force $S_{\alpha}$ stationary. By replacing $\alpha$ with $\alpha^{\prime}$ such that $S_{\alpha^{\prime}}\subseteq S_{\alpha}$ if necessary, we may assume that $\alpha\neq\beta$. Here note that $X_{\alpha}\subseteq\bar{S}_{\alpha}$ by the property (iii) of conditions of $\mathbb{P}$. Then, $X_{\alpha}\subseteq\bar{S}_{\alpha}\subseteq^{*}X_{\beta}$ and $X_{\alpha}\cap X_{\beta}=\emptyset$. Hence $X_{\alpha}$ is non-stationary in $[\lambda]^{\omega}$. Thus $\mathbb{R}_{\alpha}$ is $\omega_{1}$-stationary preserving, and so $\gamma:=\alpha$ is as desired. ∎

Now, we can prove that $\mathbb{P}$ preserves $\mathsf{MM}^{+\omega}$ by a similar argument as Beaudoin [1]:

Lemma 3.5**.**

$\mathsf{MM}^{+\omega}$* holds in $V^{\mathbb{P}}$.*

Proof.

Let $\dot{\mathbb{Q}}$ be a $\mathbb{P}$-name for an $\omega_{1}$-stationary preserving foricng notion. For each $\xi<\omega_{1}$, let $\dot{D}_{\xi}$ be a $\mathbb{P}$-name for a dense subset of $\dot{\mathbb{Q}}$, and for each $n<\omega$, let $\ddot{T}_{n}$ be a $\mathbb{P}$-name for a $\dot{\mathbb{Q}}$-name for a stationary subset of $\omega_{1}$. Take an arbitrary $p_{0}\in\mathbb{P}$. It suffices to find $p^{*}\leq p_{0}$ in $\mathbb{P}$ such that if $G$ is a $\mathbb{P}$-generic filter over $V$ with $p^{*}\in G$, then in $V[G]$ there is a filter $h\subseteq\mathbb{Q}$ with the following properties:

1. (i)

$h\cap D_{\xi}\neq\emptyset$ for any $\xi<\omega_{1}$. 2. (ii)

$\dot{T}_{n}^{h}$ is stationary in $\omega_{1}$ for all $n<\omega$.

Here $\mathbb{Q}$, $D_{\xi}$ and $\dot{T}_{n}$ denote $\dot{\mathbb{Q}}^{G}$, $\dot{D}_{\xi}^{G}$ and $\ddot{T}_{n}^{G}$, respectively.

First, we find $p^{*}$ as above. We work in $V$. We identify each $\ddot{T}_{n}$ with a $\mathbb{P}*\dot{\mathbb{Q}}$-name. Let $\dot{\gamma}$ and $\mathbb{S}$ be as in Lemma 3.4. Note that $\mathbb{S}$ is $\omega_{1}$-stationary preserving and each $\ddot{T}_{n}$ is stationary in $\omega_{1}$ in $V^{\mathbb{S}}$. Let $\dot{\mathbb{R}}$ be a $\mathbb{P}*\dot{\mathbb{Q}}$-name for $\mathbb{R}([\omega_{2}^{V}]^{\omega}\setminus\dot{X}_{\dot{\gamma}})$.

Take a sufficiently large regular cardinal $\theta$. By Fact 2.3, there are $M\in[\mathcal{H}_{\theta}]^{\omega_{1}}$ and $k\subseteq\mathbb{S}\cap M$ such that

1. (iii)

$\omega_{1}\cup\{p_{0},\dot{\mathbb{Q}},\dot{\gamma},\mathbb{S}\}\cup\{\dot{D}_{\xi}\mid\xi<\omega_{1}\}\cup\{\ddot{T}_{n}\mid n<\omega\}\subseteq M\prec\langle\mathcal{H}_{\theta},\in\rangle$, 2. (iv)

$k$ is an $M$-generic filter on $\mathbb{S}\cap M$ with $p_{0}*1_{\dot{\mathbb{Q}}}*1_{\dot{\mathbb{R}}}\in k$, 3. (v)

$\ddot{T}_{n}^{k}$ is stationary in $\omega_{1}$ for any $\xi<\mu$.

Let $\delta^{*}:=M\cap\omega_{2}\in\omega_{2}$, and let

[TABLE]

Then, $g$ is an $M$-generic filter on $\mathbb{P}\cap M$.

Note that $\sup_{p\in g}\delta_{p}=\delta^{*}$ by Lemma 3.1 (4) and the $M$-genericity of $g_{0}$. Let $p^{*}:\delta^{*}\times[\delta^{*}]^{\omega}\to 2$ be an extension of $\bigcup g$ such that $p^{*}(\alpha,x)=0$ for all $\langle\alpha,x\rangle\notin\mathop{\mathrm{dom}}\nolimits(\bigcup g)$. We claim that $p^{*}$ is as desired. For this, we use the transitive collapse of $M$. First, we make some preliminaries on it.

Let $\pi:M\to M^{\prime}$ be the transitive collapse of $M$, and let $\mathbb{P}^{\prime}$, $\dot{\mathbb{Q}}^{\prime}$, $\dot{\mathbb{R}}^{\prime}$, $\mathbb{S}^{\prime}$, $k^{\prime}$ and $g^{\prime}$ be $\pi(\mathbb{P})$, $\pi(\dot{\mathbb{Q}})$, $\pi(\dot{\mathbb{R}})$, $\pi(\mathbb{S})$, $\pi[k]$ and $\pi[g]$, respectively. Note that $\mathbb{S}^{\prime}=\mathbb{P}^{\prime}*\dot{\mathbb{Q}}^{\prime}*\dot{\mathbb{R}}^{\prime}$ in $M^{\prime}$. Moreover, $k^{\prime}$ is an $\mathbb{S}^{\prime}$-generic filter over $M^{\prime}$, and $g^{\prime}$ is the $\mathbb{P}^{\prime}$-generic filter over $M^{\prime}$ naturally obtained from $k^{\prime}$. Let $h^{\prime}$ be the $(\dot{\mathbb{Q}}^{\prime})^{g^{\prime}}$-generic filter over $M^{\prime}[g^{\prime}]$ naturally obtained from $k^{\prime}$, and let $i^{\prime}$ be the $(\dot{\mathbb{R}}^{\prime})^{g^{\prime}*h^{\prime}}$-generic filter over $M^{\prime}[g^{\prime}*h^{\prime}]$ naturally obtained from $k^{\prime}$.

Now, we start to prove that $p^{*}$ is as desired. First, we prove that $p^{*}\in\mathbb{P}$. We only check that $X_{p^{*},\gamma}$ is non-stationary in $[\delta^{*}]^{\omega}$ for some $\gamma<\delta^{*}$. The other properties are easily checked.

First of all, note that $\pi\!\upharpoonright\!(\mathcal{H}_{\omega_{2}}\cap M)$ is the identity map since $\mathcal{H}_{\omega_{2}}\cap M$ is transitive and that $\pi(\omega_{2})=\delta^{*}$. Let $\gamma:=\pi(\dot{\gamma})^{g^{\prime}*h^{\prime}}<\pi(\omega_{2})=\delta^{*}$. Then $\mathrm{range}(\bigcup i^{\prime})$ is a club subset of $[\delta^{*}]^{\omega}$ which does not intersect $\bigcup_{p^{\prime}\in g^{\prime}}X_{p^{\prime},\gamma}=\bigcup_{p\in g}\pi(X_{p,\gamma})$. Here note that $X_{p,\gamma}\in\mathcal{H}_{\omega_{2}}\cap M$ for all $p\in g$ by the property (ii) of conditions in $\mathbb{P}$. So $\bigcup_{p\in g}\pi(X_{p,\gamma})=\bigcup_{p\in g}X_{p,\gamma}=X_{p^{*},\gamma}$. Hence $X_{p^{*},\gamma}$ is non-stationary in $[\delta^{*}]^{\omega}$

We have shown that $p^{*}\in\mathbb{P}$. Note that $p^{*}$ is a lower bound of $g$. Then $p^{*}\leq p$ since $p\in g$ by (iv). Suppose $G$ is a $\mathbb{P}$-generic filter over $V$ with $p^{*}\in G$. Working in $V[G]$, we find a filter $h\subseteq\mathbb{Q}$ satisfying (i) and (ii).

Let $M[G]$ denote the collection of $\dot{a}^{G}$ for all $\mathbb{P}$-names $\dot{a}\in M$, and define $\hat{\pi}:M[G]\to M^{\prime}[g^{\prime}]$ by $\hat{\pi}(\dot{a}^{G}):=\pi(\dot{a})^{g^{\prime}}$. It is easy to see that $\hat{\pi}$ coincides with the transitive collapse of $M[G]$ and that $\hat{\pi}$ extends $\pi$. Let $h$ be the filter on $\mathbb{Q}$ generated by $\hat{\pi}^{-1}[h^{\prime}]$. Then $h$ satisfies (i) since $D_{\xi}\in M[G]$ and $h^{\prime}\cap\hat{\pi}(D_{\xi})\neq\emptyset$ for all $\xi<\omega_{1}$. As for (ii), it is easy to see that $\dot{T}_{n}^{h}=\ddot{T}_{n}^{k}$ for each $n<\omega$. Then, $h$ satisfies (ii) by (v). ∎