Properness under closed forcing

TL;DR

This paper explores how certain proper posets can become improper after applying specific $ ext{κ}$-closed forcing extensions, revealing nuanced behavior of properness under forcing.

Contribution

It provides explicit examples of proper posets that lose their properness in some $ ext{κ}$-closed forcing extensions, highlighting limitations of properness preservation.

Findings

Proper posets can become improper after $ ext{κ}$-closed forcing.

Examples demonstrate properness is not always preserved under certain extensions.

Properness behavior varies with different forcing extensions.

Abstract

For every uncountable regular $\kappa$, we give two examples of proper posets which turn improper in some $\kappa$-closed forcing extension.

Equations

$${⊩}_{\mathbb{Q}}\text{“Pˇ is not proper.”}$$

$$[a,x]:=\{y\in [x{]}^{\omega }\mid a\subseteq y\}\subseteq \Sigma (M).$$

$$C(f)=\{x\in [\kappa {]}^{\omega }\mid \text{x is closed under f}\}\subseteq C.$$

$$\tilde{C}:=C(f)\setminus \{{\mathrm{cl}}_f(a)\mid a\in [\kappa {]}^{<\omega }\},$$

$$\mathcal{E}=\{M\in [H(\theta ){]}^{\omega }\mid \mathbb{Q},\dot{C}\in M\prec H(\theta )\}.$$

$${⊩}_{\mathbb{Q}}\text{“{x∈C˙∣α˙nM∈x}⊆(C~nM)ˇ.”}$$

$${⊩}_{\mathbb{Q}}\text{“α˙nM∈a˙nM∈[y˙nM]<ω∧y˙nM∈(CnM∖C~nM)ˇ∧[a˙nM,y˙nM]∩(C~nM)ˇ=∅,”}$$

$${⊩}_{\mathbb{Q}}\text{“[a˙nM,y˙nM]∩C˙=∅.”}$$

$$p_M{⊩}_{\mathbb{Q}}\text{“[(anM)ˇ,(ynM)ˇ ]∩C˙=∅.”}$$

$$p_M{⊩}_{\mathbb{Q}}\text{“(Σ(M))ˇ∩C˙=∅.”}$$

$$p{⊩}_{\mathbb{Q}}\text{“{M∈Eˇ∣pM∈G˙} is stationary in [(H(θ)V)ˇ ]ω, ”}$$

$${⊩}_{\mathbb{Q}}\text{“D˙ is a club subset of Eˇ and ∀M∈D˙[pM∈/G˙].”}$$

$$p{⊩}_{\mathbb{Q}}\text{“Pˇ is not proper.”}$$

$$\mathcal{F}:=\{M\in \mathcal{E}\mid M\cap \kappa \in {\dot{C}}_G\}.$$

$${\mathcal{D}}_{\alpha }:=\{q\in \mathbb{P}\mid \exists \xi \in \mathrm{dom}(q)[\xi \ge \alpha \wedge q(\xi )\in \mathcal{F}]\}.$$

$$\mathcal{S}:=\{N\in [H(\lambda ){]}^{\omega }\mid \mathbb{P},({\mathcal{D}}_{\alpha }{)}_{\alpha <{\omega }_1}\in N\prec H(\lambda )\wedge N\cap (H(\theta ){)}^V\in S\}.$$

$$\{\xi <\delta \mid q(\xi )\cap \kappa \in {\dot{C}}_G\}$$

$$q(\xi )\cap \kappa \in \Sigma (M)$$

Full text

The author was supported by Grant-in-Aid for Scientific Research (C) (No. 18K03394) from JSPS

1. Introduction

In this paper we prove the following theorem:

Theorem 1**.**

Let $\kappa$ be an uncountable regular cardinal. Then there exists a $\kappa$-closed poset $\mathbb{Q}$ and a proper poset $\mathbb{P}$ such that

[TABLE]

This theorem negatively answers the question raised by Karagila [4], which asks if there exists a sufficiently large cardinal $\kappa$ such that any proper poset remains proper after any $\kappa$-closed forcing. The motivation of this question is discussed in a recent article by Asperó and Karagila [1].

In fact, we give two examples witnessing the conclusion of Theorem 1. The first example is a rather simple mimic of the known example of a pair of proper posets whose product is improper, given by Shelah [7, XVII Observation 2.12, p.826]. This example is briefly mentioned in [1]. The second example involves the class of posets introduced by Moore [6] in connection with the Mapping Reflection Principle ($\mathrm{MRP}$) he introduced in the same paper. While the second example requires a longer argument, it has the advantage that $\mathbb{P}$ is taken to be totally proper.

2. The First Example

Let $T$ be the complete binary tree of height $\kappa$. Let $\mathbb{Q}=T$ with the reversed ordering. Clearly $\mathbb{Q}$ is $\kappa$-closed. Let $\mathbb{P}_{0}=\mathrm{Add}(\omega)$ and $\dot{\mathbb{P}}_{1}=\mathrm{Col}(\omega_{1},2^{\kappa})$ in $V^{\mathbb{P}_{0}}$. $\dot{\mathbb{P}}_{1}$ is $\sigma$-closed in $V^{\mathbb{P}_{0}}$. Note that, no new cofinal branches are added to $T$ by forcing with $\mathbb{P}_{0}*\dot{\mathbb{P}}_{1}$, by a well-known argument first proposed by Mitchell [5]. Let $\dot{C}$ be a $(\mathbb{P}_{0}*\dot{\mathbb{P}}_{1})$-name for a cofinal subset of $\kappa$ of order type $\omega_{1}$, and let $T\upharpoonright\dot{C}$ denote the subset of $T$ consisting of all nodes of level in $\dot{C}$ (defined in $V^{\mathbb{P}_{0}*\dot{\mathbb{P}}_{1}}$). Note that, in $V^{\mathbb{P}_{0}*\dot{\mathbb{P}}_{1}}$, $T\upharpoonright\dot{C}$ forms a tree of size and height $\omega_{1}$, and since every cofinal branch through $T\upharpoonright\dot{C}$ generates one through $T$, which is in $V$, the number of cofinal branches through $T\upharpoonright\dot{C}$ is also $\omega_{1}$. Now let $\dot{\mathbb{P}}_{2}$ be a $(\mathbb{P}_{0}*\dot{\mathbb{P}}_{1})$-name for the c.c.c. poset specializing $T\upharpoonright\dot{C}$, as described in Baumgartner [2, §7], and set $\mathbb{P}:=\mathbb{P}_{0}*\dot{\mathbb{P}}_{1}*\dot{\mathbb{P}}_{2}$. $\mathbb{P}$ is a three step iteration of c.c.c., $\sigma$-closed and c.c.c. posets, and thus is proper. Note that, since $T\upharpoonright\dot{C}$ is specialized in $V^{\mathbb{P}}$, $\omega_{1}$ must be collapsed in any further extension where new cofinal branches through $T\upharpoonright\dot{C}$ are added. Since forcing with $\mathbb{Q}$ over $V$ adds a new cofinal branch through $T$, forcing with $\mathbb{Q}\times\mathbb{P}$ adds a cofinal branch through $T\upharpoonright\dot{C}$ which is not in $V^{\mathbb{P}}$. Therefore $\mathbb{Q}\times\mathbb{P}$ collapses $\omega_{1}$, and thus is improper. This shows that $\mathbb{P}$ is improper in $V^{\mathbb{Q}}$.

3. The Second Example

First let us review some relevant definitions and facts we will use in construction of our second example. Let us start with the notion of total properness.

Definition 2**.**

Let $\mathbb{P}$ be a poset, and $N$ a countable $\in$-model of a suitable fragment of $\mathrm{ZFC}$ which contains $\mathbb{P}$.

1. (1)

$p\in\mathbb{P}$ is said to be $(N,\mathbb{P})$-generic if for every dense subset $D\in N$ of $\mathbb{P}$, $D\cap N$ is predense below $p$. 2. (2)

$p\in\mathbb{P}$ is said to be totally $(N,\mathbb{P})$-generic if for every dense subset $D\in N$ of $\mathbb{P}$, $p$ extends some element of $D\cap N$.

Note that $\mathbb{P}$ is proper iff for every sufficiently large regular $\theta$, every countable $N\prec H(\theta)$ containing $\mathbb{P}$ and every $p\in N\cap\mathbb{P}$, there exists an $(N,\mathbb{P})$-generic condition of $\mathbb{P}$ which extends $p$.

Definition 3**.**

A poset $\mathbb{P}$ is said to be totally proper if for every sufficiently large regular $\theta$, every countable $N\prec H(\theta)$ containing $\mathbb{P}$ and every $p\in N\cap\mathbb{P}$, there exists a totally $(N,\mathbb{P})$-generic condition of $\mathbb{P}$ which extends $p$.

The notion of total properness has the following simple characterization.

Theorem 4** (Eisworth-Roitman[3]).**

A poset $\mathbb{P}$ is totally proper iff $\mathbb{P}$ is proper and $\sigma$-Baire111$\sigma$-Baire posets are sometimes referred to as $\sigma$-distributive posets..∎

Theorem 4 can be proved using the following lemma, which we will use later.

Lemma 5**.**

Suppose $\mathbb{P}$ is $\sigma$-Baire and $N$ is countable. Then every $(N,\mathbb{P})$-generic condition can be extended to a totally $(N,\mathbb{P})$-generic condition.∎

Next let us review the class of posets first introduced by Moore [6]. They were originally designed to prove the Mapping Reflection Principle ($\mathrm{MRP}$) from the Proper Forcing Axiom ($\mathrm{PFA}$). In our second example, $\mathbb{P}$ will be taken from this class.

Definition 6** (Moore [6]).**

Let $\theta$ be a regular uncountable cardinal, and $X\in H(\theta)$. For a club subset $\mathcal{E}$ of $\{M\in[H(\theta)]^{\omega}\mid M\prec H(\theta)\}$, a function $\Sigma:\mathcal{E}\to\mathcal{P}([X]^{\omega})$ is said to be an open stationary set mapping if for every $M\in\mathcal{E}$,

1. (1)

$\Sigma(M)$ is $M$-stationary, that is, $\Sigma(M)\cap C\cap M\not=\emptyset$ for every club subset $C\in M$ of $[X]^{\omega}$, and 2. (2)

$\Sigma(M)$ is open in the Ellentuck topology, that is, for every $x\in\Sigma(M)$ there exists a finite $a\subseteq x$ such that

[TABLE]

For an open stationary set mapping $\Sigma:\mathcal{E}\to\mathcal{P}([X]^{\omega})$, the poset $\mathcal{P}_{\Sigma}$ is defined as follows: $\mathcal{P}_{\Sigma}$ consists of the functions of the form $q:\alpha+1\to\mathcal{E}$ for some $\alpha<\omega_{1}$ such that

1. (1)

$q(\gamma)\in q(\gamma+1)$ for every $\gamma<\alpha$, 2. (2)

$q(\gamma)=\bigcup_{\xi<\gamma}q(\xi)$ for every limit $\gamma\leq\alpha$, and 3. (3)

for every limit $\gamma\leq\alpha$, there exists $\nu<\gamma$ such that $q(\xi)\cap X\in\Sigma(q(\gamma))$ for all $\xi$ satisfying $\nu<\xi<\gamma$.

$\mathcal{P}_{\Sigma}$ is ordered by initial segment.

The following lemma is the heart of Moore’s proof of $\mathrm{MRP}$ from $\mathrm{PFA}$.

Lemma 7** (Moore [6]).**

$\mathcal{P}_{\Sigma}$ is totally proper.∎

The following are density lemmata about $\mathcal{P}_{\Sigma}$, which we will use later. For $q\in\mathcal{P}_{\Sigma}$, we write $q(\mathrm{dom}(q)-1)$ as $\mathrm{last}(q)$.

Lemma 8** (Moore [6]).**

Let $\Sigma:\mathcal{E}\to\mathcal{P}([X]^{\omega})$ be an open stationary set mapping, where $\mathcal{E}$ is a club subset of $\{M\in[H(\theta)]^{\omega}\mid M\prec H(\theta)\}$ for an uncountable regular cardinal $\theta$.

1. (1)

For every $\alpha<\omega_{1}$, $D_{\alpha}:=\{q\in\mathcal{P}_{\Sigma}\mid\alpha\in\mathrm{dom}(q)\}$ is dense in $\mathcal{P}_{\Sigma}$. 2. (2)

For every $x\in H(\theta)$, $E_{x}:=\{q\in\mathcal{P}_{\Sigma}\mid x\in\mathrm{last}(q)\}$ is dense in $\mathcal{P}_{\Sigma}$.∎

We also use the following lemma in our construction.

Lemma 9**.**

Let $\kappa$ be an uncountable regular cardinal. For every club subset $C$ of $[\kappa]^{\omega}$, there exists a club subset $\tilde{C}$ of $C$ such that

1. (a)

$\tilde{C}$ is closed in the Ellentuck topology. 2. (b)

$C\setminus\tilde{C}$ is weakly unbounded, that is, for every $a\in[\kappa]^{<\omega}$ there exists $x\in C\setminus\tilde{C}$ such that $a\subseteq x$.

Proof.

Let $f:{}^{<\omega}\kappa\to\kappa$ be such that

[TABLE]

Let

[TABLE]

where $\mathrm{cl}_{f}(a)$ denotes the closure of $a$ by $f$. Since every union of a strictly $\subseteq$-increasing $\omega$-sequence of elements of $C(f)$ is in $\tilde{C}$, it easily follows that $\tilde{C}$ is a club subset of $C$. If $x\in[\kappa]^{\omega}$ is an accumulation point of $\tilde{C}$ in Ellentuck topology, $x$ must be closed under $f$ and also cannot be finitely generated by $f$, and thus is in $\tilde{C}$. This shows (a). Since $a\subseteq\mathrm{cl}_{f}(a)\in C\setminus\tilde{C}$ holds for every $a\in[\kappa]^{<\omega}$, we have (b).∎∎

Now we start to describe our construction of the second example. Suppose a regular uncountable cardinal $\kappa$ is given. Let $\mathcal{C}$ be the set of club subsets of $[\kappa]^{\omega}$, and let $\mathbb{Q}=Col(\kappa,|\mathcal{C}|)$. Clearly $\mathbb{Q}$ is $\kappa$-closed, and $|\mathcal{C}|=\kappa$ holds in $V^{\mathbb{Q}}$. Let $\dot{C}$ be the $\mathbb{Q}$-name for a diagonal intersection of the members of $\mathcal{C}$. $\dot{C}$ is a club subset of $[\kappa]^{\omega}$ in $V^{\mathbb{Q}}$.

Let $\theta$ be a sufficiently large regular cardinal, and let

[TABLE]

Fix an arbitrary $M\in\mathcal{E}$. Pick a totally $(M,\mathbb{Q})$-generic condition $p_{M}\in\mathbb{Q}$. Let $\{C^{M}_{n}\mid n<\omega\}$ enumerate $M\cap\mathcal{C}$. Note that, for each $n<\omega$, we may assume that $\tilde{C}^{M}_{n}\in M\cap\mathcal{C}$, where $\tilde{C}^{M}_{n}$ is made from $C^{M}_{n}$ as in Lemma 9. By the definition of $\dot{C}$, for each $n<\omega$ there exists a $\mathbb{Q}$-name $\dot{\alpha}^{M}_{n}\in M$ (for an ordinal below $\kappa$) such that

[TABLE]

By (a) and (b) of Lemma 9, there exists a $\mathbb{Q}$-name $\dot{y}^{M}_{n}$ and $\dot{a}^{M}_{n}$ both in $M$ such that

[TABLE]

By (1) and (2) we have

[TABLE]

By the total genericity of $p_{M}$, for each $n<\omega$ there exists $y^{M}_{n}\in C^{M}_{n}\cap M$ and $a^{M}_{n}\in[y^{M}_{n}]^{<\omega}$ such that

[TABLE]

Now let us set $\Sigma(M):=\bigcup_{n<\omega}[a^{M}_{n},y^{M}_{n}]$. It is easy to check that $\Sigma$ is an open stationary set mapping ($\Sigma(M)$ is clearly open, and is $M$-stationary since $y^{M}_{n}\in\Sigma(M)\cap C^{M}_{n}\cap M$ for every $n<\omega$). By (3) we have

[TABLE]

Now let $\mathbb{P}:=\mathcal{P}_{\Sigma}$. $\mathbb{P}$ is totally proper by Lemma 7.

Lemma 10**.**

There exists $p\in\mathbb{Q}$ such that

[TABLE]

where $\dot{G}$ is the canonical $\mathbb{Q}$-name for $\mathbb{Q}$-generic filter.

Proof.

Suppose otherwise. Then there exists a $\mathbb{Q}$-name $\dot{D}$ such that

[TABLE]

Let $\lambda$ be a sufficiently large regular cardinal, and let $N$ be a countable elementary submodel of $H(\lambda)$ such that $\dot{D}\in N$ and $M:=N\cap H(\theta)\in\mathcal{E}$. Now let $G$ be a $\mathbb{Q}$-generic filter over $V$ such that $p_{M}\in G$. Then by genericity of $p_{M}$ it holds that $N[G]\cap(H(\theta))^{V}=N\cap(H(\theta))^{V}=M$. On the other hand, since $\dot{D}\in N$ and $\dot{D}$ is forced to be a club, it holds that $N[G]\cap(H(\theta))^{V}\in\dot{D}_{G}$. Thus it follows that $M\in\dot{D}_{G}$, and $p_{M}\notin G$ should be the case. This is a contradiction.∎∎

Now let $p$ be as in Lemma 10. We will show that

[TABLE]

Note that it is enough for our goal: we may rename $\mathbb{Q}$ below $p$ as $\mathbb{Q}$, or use the weak homogeneity of $\mathbb{Q}$.

To this end, let $G$ be a $\mathbb{Q}$-generic filter over $V$ such that $p\in G$. Work in $V[G]$. Let

[TABLE]

Since $\dot{C}_{G}$ is a club subset of $[\kappa]^{\omega}$, $\mathcal{F}$ is a club subset of $\mathcal{E}\subseteq[(H(\theta))^{V}]^{\omega}$.

Now for each $\alpha<\omega_{1}$, let

[TABLE]

It is clear that $\mathcal{D}_{\alpha}$ is a dense subset of $\mathbb{P}$ (by Lemma 8(1), and the unboundedness of $\mathcal{F}$; note that any $q\in\mathbb{P}$ can be extended by adding any member of $\mathcal{E}$ which contains $\mathrm{last}(q)$ as an element).

Now let $S:=\{M\in\mathcal{E}\mid p_{M}\in G\}$. By our choice of $p$, $S$ is a stationary subset of $[(H(\theta))^{V}]^{\omega}$.

Now let $\lambda$ be a sufficiently large regular caridnal, and set

[TABLE]

Then $\mathcal{S}$ is stationary in $[H(\lambda)]^{\omega}$. Pick any $N\in\mathcal{S}$. Set $M:=N\cap(H(\theta))^{V}\in S$ and $\delta:=N\cap\omega_{1}=M\cap\omega_{1}$. The following claim is enough to show (5).

**(Claim) **.

There are no $(N,\mathbb{P})$-generic conditions.

Proof.

Suppose there is an $(N,\mathbb{P})$-generic condition. Since $\mathbb{P}$ remains $\sigma$-Baire in $V[G]$, by Lemma 5 such a condition can be extended to a totally $(N,\mathbb{P})$-generic condition $q$. By the total genericity of $q$, Lemma 8(1)(2), the facts that the domain of every condition of $\mathbb{P}\cap N$ is less than $\delta$, and that $\mathcal{D}_{\alpha}\in N$ for every $\alpha<\delta$, we may assume that

1. (a)

$\mathrm{dom}(q)=\delta+1$, 2. (b)

$\mathrm{last}(q)=q(\delta)=M$ and 3. (c)

$\{\xi<\delta\mid q(\xi)\in\mathcal{F}\}$ is unbounded in $\delta$.

By (c) we have

[TABLE]

On the other hand, by (b) there exists $\nu<\delta$ such that

[TABLE]

But since $M\in S$, $p_{M}\in G$ holds, and thus by (4) we have $\Sigma(M)\cap\dot{C}_{G}=\emptyset$. This is a contradiction.∎∎

This finishes our construction.

Note that our second example is more delicate than the first one in the following sense: although the properness of $\mathbb{P}$ is destroyed by forcing over $\mathbb{Q}$, the product $\mathbb{Q}\times\mathbb{P}$ remains proper unlike the first example, because $\mathbb{P}$ is $\sigma$-Baire and therefore $\mathbb{Q}$ remains $\sigma$-closed by forcing over $\mathbb{P}$.