Set theory seminar -Forcing axioms and inner models III

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(At Randall’s request, this entry will be more detailed than usual.)

Remark 1. $\omega_1$ is club in ${\mathcal P}_{\omega_1}(\omega_1)$ , so any $S\subseteq\omega_1$ is stationary as a subset of $\omega_1$ iff it is stationary as a subset of ${\mathcal P}_{\omega_1}(\omega_1)$ . It follows that proper forcing preserves stationary subsets of $\omega_1$ .

Remark 2. Proper forcing extensions satisfy the countable covering property with respect to $V$ , namely, if ${\mathbb P}$ is proper, then any countable set of ordinals in $V^{\mathbb P}$ is contained in a countable set of ordinals in $V$ . We won’t prove this for now, but the argument for Axiom A posets resembles the general case reasonably:

Given a name $\dot X$ for a countable set of ordinals in the extension, find an appropriate regular $\theta$ and consider a countable elementary $N\prec H_\theta$ containing $\dot X$ , ${\mathbb P}$ , and any other relevant parameters. One can then produce a sequence $(p_n)_{n\in\omega}$ such that

Each $p_i$ is in $N$ .
$p_{i+1}\le_i p_i$ .
$p_i\in D_i$ , where $(D_n)_{n\in\omega}$ enumerates the dense subsets of ${\mathbb P}$ in $N$ .

Let $p\le_i p_i$ for all $i$ . Then $p\Vdash \dot X\subseteq N$ , so $N\cap{\sf ORD}$ is a countable set of ordinals in $V$ containing $X$ in $V^{\mathbb P}$ . A density argument completes the proof.

Woodin calls a poset ${\mathbb P}$ weakly proper if the countable covering property holds between $V$ and $V^{\mathbb P}$ . Not every weakly proper forcing is proper, but the countable covering property is a very useful consequence of properness. On the other hand, standard examples of stationary set preserving posets that are not proper, like Prikry forcing (changing the cofinality of a measurable cardinal $\kappa$ to $\omega$ without adding bounded subsets of $\kappa$ ) or Namba forcing (changing the cofinality of $\omega_2$ to $\omega$ while preserving $\omega_1$ are not weakly proper, and account for some of the usefulness of ${\sf MM}$ over ${\sf PFA}$ .

The following is obvious:

Fact. Assume ${\mathbb P}$ is weakly proper. Then either ${\mathbb P}$ adds no new $\omega$ -sequences of ordinals, or else it adds a real.

The relation between the reals and the $\omega$ -sequences of ordinals in the presence of strong forcing axioms like ${\sf PFA}$ is a common theme I am exploring through these talks.

Preserving forcing axioms.

To motivate the main line of results (relating models of forcing axioms to their inner models) I want to discuss some preservation results.

The first is fairly easy.

Fact. ${\sf BPFA}$ is preserved by any proper forcing that does not add subsets of $\omega_1$ .

For if ${\sf BPFA}$ holds, ${\mathbb P}$ is such a poset, and $\dot{\mathbb Q}$ is a ${\mathbb P}$ -name for a proper poset, then $H_{\omega_2}^V=H_{\omega_2}^{V^{\mathbb P}}$ and ${\mathbb P}*\dot {\mathbb Q}$ is proper, so $H_{\omega_2}^{V^{\mathbb P}}\prec_1 V^{{\mathbb P}*\dot{\mathbb Q}}$ .

Trivial as this fact is, removing the assumption of properness of ${\mathbb P}$ makes matters much more subtle. Closely related to this is the following question:

Open question 1. Assume $V$ is a forcing extension of $L$ and ${\sf BPFA}$ holds. Does $L({\mathcal P}(\omega_1))\models{\sf BPFA}$ ?

Moore has shown that ${\sf BPFA}$ implies that $L({\mathbb P}(\omega_1))$ is a model of choice, so in the situation above, it is a forcing extension of $L$ . Even if $V$ is an extension of $L$ by proper forcing (even if $V$ is an extension of $L$ by the standard iteration forcing ${\sf BPFA}$ ), I do not see that it is an extension of $L({\mathcal P}(\omega_1))$ by proper forcing, so the obvious argument above does not seem to apply.

Theorem (König, Yoshinobu). ${\sf PFA}$ is preserved by $<\omega_2$ -closed forcing.

As in the case of ${\sf BPFA}$ , this establishes preservation under certain forcing posets that do not change $H_{\omega_2}$ . In fact, $<\omega_2$ -closed forcing adds no new $\omega_1$ -sequences of ordinals to the universe. Something is required here, in any case, in light of the following result:

Theorem (Caicedo, Velickovic). If $W$ is an outer model of $V$ with the same $\omega_2$ , and ${\sf BPFA}$ holds in both $V$ and $W$ , then ${\mathcal P}(\omega_1)^V={\mathcal P}(\omega_1)^W$ .

So, if we have a forcing that preserves ${\sf PFA}$ but adds a subset of $\omega_1$ , then it must collapse $\omega_2$ . The cardinal $\omega_2$ seems to play a key role in the structure of models of ${\sf PFA}$ . The only known methods of forcing ${\sf PFA}$ collapse a supercompact to $\omega_2$ , so we expect that, in the presence of ${\sf PFA},$ $\omega_2$ has large cardinal properties in inner models. This leads me to believe that the following question should have a negative answer. (I believe the question is due to Miguel Angel Mota.)

Open question 2. Assume that ${\sf PFA}$ holds and there are no inaccessibles. Let $c$ be a Cohen real over $V$ . Is there an outer model of $V[c]$ where ${\sf PFA}$ holds?

That ${\sf PFA}$ fails in $V[c]$ itself follows from Shelah’s result that adding a Cohen real adds a Suslin tree, but by the Caicedo-Velickovic theorem above, we see that in fact any forcing that adds a real or a subset of $\omega_1$ without collapsing $\omega_2$ destroys ${\sf PFA}$ .

The question assumes that there are no inaccessible cardinals in the universe to avoid “cop out” solutions where ${\sf PFA}$ is simply forced again by using a supercompact present in the universe, or slightly more subtly, by perhaps resurrecting a former supercompact and then forcing with it. Of course, no such resurrection is possible without inaccessibles in the universe.

(I don’t know of any progress on this question. Any information you may have, I would be very interested in hearing about it.)

Following with the theme that ${\sf MM}$ is perhaps the most natural strong forcing axiom to study, rather than proving the König-Yoshinobu theorem, I want to prove the following result:

Theorem (Larson). ${\sf MM}$ is preserved by $<\omega_2$ -directed closed forcing.

Recall:

Definition. A poset ${\mathbb P}$ is $<\kappa$ -directed closed iff any directed $D\in {\mathcal P}_\kappa({\mathbb P})$ (i.e., for any $p,q\in D$ there is $r\in D$ with $r\le p$ and $r\le q$ ) admits a lower bound in ${\mathbb P}$ .

Clearly, any such forcing is $<\kappa$ -closed, which requires $D$ to be a decreasing sequence, but being directed closed is more restrictive.

Proof. Assume ${\sf MM}$ and let ${\mathbb P}$ be $<\omega_2$ -directed closed. Let $\dot{\mathbb Q}$ be a ${\mathbb P}$ -name for a stationary set preserving forcing. Then ${\mathbb P}*\dot{\mathbb Q}$ is stationary set preserving as well. Fix also a ${\mathbb P}$ -name $(\tau_\xi:\xi<\omega_1)$ for a sequence of $\omega_1$ many dense subsets of $\dot{\mathbb Q}.$

For $\xi<\omega_1$ , let $D_\xi=\{(p,\dot q): p\Vdash\dot q\in\tau_\xi\}$ and let $E_\xi$ be the first-coordinate projection of $D_\xi$ .

By ${\sf MM}$ , there is a filter $K\subseteq {\mathbb P}*\dot{\mathbb Q}$ meeting all the $D_\xi$ . Let $G$ be its first coordinate projection. $G$ is a directed subset of ${\mathbb P}$ , and we can find a directed subset of size at most $\omega_1$ meeting all the $E_\xi$ . Since ${\mathbb P}$ is $<\omega_2$ -directed closed, we can find a lower bound $p$ for $G$ . Thus, $p\Vdash\mbox{``}\{\dot q:\exists p'\ge p\,(p',q)\in\check G\}\mbox{ is a filter meeting all the }\tau_\xi.\mbox{''}$ ${\sf QED}$

Remark 3. König and Yoshinobu have shown that for any $\lambda$ there is a $<\lambda$ -closed forcing that destroys ${\sf MM}$ . Their argument generalizes the fact that under ${\sf BPFA}$ there are no weak Kurepa trees and in fact every tree of height and size $\omega_1$ is special; we will review this result later. It also makes essential use of Namba forcing.

Remark 4. The above remark indicates that in a sense the König-Yoshinobu preservation theorem is as strong as possible. There is another sense in which it cannot be strengthened either: Given an ordinal $\alpha$ , recall that a poset ${\mathbb P}$ is strongly $\alpha$ -game closed or $\alpha$ -strategically closed iff Nonempty has a winning strategy for the game where Empty and Nonempty alternate playing conditions $p_0\ge p_1\ge\dots$ in ${\mathbb P}$ for $\alpha$ stages and Nonempty wins iff there is a lower bound to the sequence produced this way. This is a weaker notion than $(<|\alpha|^+)$ -closed, and a useful substitute in many instances. However, ${\sf PFA}$ is not preserved by strongly $\omega_1$ -game closed forcing. For example, $\square_{\omega_1}$ can be added with such a forcing.

Remark 5. Similarly, one can add a $\square(\omega_2)$ -sequence and then destroy it and the whole extension adds no new $\omega_1$ -sequences of ordinals and preserves ${\sf PFA}$ , but ${\sf PFA}$ fails in the intermediate extension, so preservation is a subtler matter and certainly does not depend on the preservation of ${\sf ORD}^{\omega_1}$ alone.

Rather than proving the König-Yoshinobu preservation theorem, let me give a quick sketch, emphasizing that the argument is very similar to the proof of Larson’s result given above. Now we consider a $<\omega_2$ -closed poset ${\mathbb P}$ , a ${\mathbb P}$ -name $\dot{\mathbb Q}$ for a proper poset, and a ${\mathbb P}$ -name $(\tau_\alpha:\alpha<\omega_1)$ for a sequence of dense subsets of (the interpretation of) $\dot{\mathbb Q}$ . As in Larson’s proof we can find a directed $G\subseteq{\mathbb P}$ of size at most $\omega_1$ meeting the projections of the corresponding dense sets $D_\xi$ . However, the assumption on ${\mathbb P}$ does not suffice to pick up a lower bound of $G$ . Rather, a new poset ${\mathbb R}$ (in the extension $V^{{\mathbb P}*\dot{\mathbb Q}}$ ) is considered, that adds with countable conditions a decreasing $\omega_1$ -sequence through $G$ . The countable covering property of proper posets is used to see that ${\mathbb R}$ is $\sigma$ -closed. A further use of ${\sf PFA}$ allows us then to find an $\omega_1$ -sequence $(p_\xi:\xi<\omega_1)$ through $G$ such that for each $\xi$ there is some $\dot q_\xi$ with $(p_\xi,\dot q_\xi)\in D_\xi=\{(p,\dot q):p\Vdash\dot q\in\tau_\xi\}$ . The closure of ${\mathbb P}$ now allows us to pick a lower bound for this sequence, from which the proof is concluded as before.

Update: Sean Cox has developed a nice framework that explains all typical preservation theorems for forcing axioms in terms of the existence of certain generic elementary embeddings. See Theorem 20 in Forcing axioms, approachability at $\omega_2$ , and stationary set reflection, arXiv:1807.06129.

Not exactly a preservation result, but in the same spirit, I want to conclude by mentioning a result of Todorcevic that indicates that considering $\omega_1$ -sequences of ordinals or at least subsets of $\omega_1$ is not completely arbitrary in the setting of ${\sf PFA}$ .

Theorem (Todorcevic). Assume ${\sf PFA}$ and let ${\mathbb P}$ be a poset that adds a subset of $\omega_1$ . Then either ${\mathbb P}$ adds a reals, or else it collapses $\omega_2$ .

The argument is not too complicated, but requires a good understanding of trees on $\omega_1$ under forcing axioms. I now proceed to the beginning of this analysis.

Definition. Let $T$ be a tree of height and size $\omega_1$ . Then $T$ is special (in the restricted sense) iff $T$ is a countable union of antichains.

I won’t prove this, but the above is equivalent to the existence of an order preserving embedding $j:T\to{\mathbb R}$ of the tree into the real (or even the rational) numbers.

Notice that if $T$ is special, then it has no uncountable branches in any outer model, since any uncountable subset of $T$ must meet one of the antichains in at least two points. (For the other version, a branch would induce a subset of ${\mathbb R}$ of order type $\omega_1$ or $\omega_1^*$ , which is impossible.)

Special trees play a key role in Todorcevic’s argument. One begins by showing the following:

Theorem. Assume ${\sf MA}_{\omega_1}$ . Then every tree of height and size $\omega_1$ without uncountable branches is special.

Proof. Let $S$ be such a tree. We show that there is a ccc poset that specializes $S$ . Begin with ${\mathbb P}(S)$ , the collection of finite antichains of $S$ . We will argue that this is ccc. Now consider the product with finite support of countably many copies of ${\mathbb P}(S)$ , call it ${\mathbb P}^\omega(S)$ . By Martin’s axiom, this poset is ccc as well. Considering the dense sets $D_t=\{(p_1,\dots,p_k)\in{\mathbb P}^\omega(S):t\in\bigcup_i p_i\}$ for $t\in S$ , it follows from Martin’s axiom that there is a partition of $S$ into countably many antichains, as required.

To see that ${\mathbb P}(S)$ is ccc, fix a uniform ultrafilter ${\mathcal U}$ on $\omega_1$ and assume otherwise, so there is an uncountable antichain $\{r_\alpha:\alpha<\omega_1\}$ through ${\mathbb P}(S)$ . Each $r_\alpha$ is a finite antichain through $S$ . That they constitute an antichain in ${\mathbb P}(S)$ means that for any $\alpha<\beta$ one of the elements of $r_\alpha$ is comparable to one of the elements of $r_\beta$ . By the $\Delta$ -system lemma, we may assume all the $r_\alpha$ have the same size $n$ , and are pairwise disjoint. Write $r_\alpha=\{r_\alpha^0,\dots,r_\alpha^{n-1}\}$ . Set $A_\alpha(i,j)=\{\beta>\alpha: r_\alpha^i<r_\beta^j\}$ for $\alpha<\omega_1$ , $i,j<n$ . For each $\alpha$ , all but countably many countable ordinals are in some $A_\alpha(i,j)$ , so there are $i_\alpha,j_\alpha$ such that $A_\alpha(i_\alpha,j_\alpha)\in{\mathcal U}$ . We can fix an uncountable set $B$ and numbers $i,j$ such that $(i_\alpha,j_\alpha)=(i,j)$ for all $\alpha\in B$ .

Consider $\alpha<\beta$ , both in $B$ . Then there is some $\gamma$ in $A_\alpha(i,j)\cap A_\beta(i,j)$ , since in fact this is a set in ${\mathcal U}$ . But then both $r_\alpha^i$ and $r_\beta^i$ are below $r_\gamma^j$ so they are in fact comparable. It follows that $\{r_\alpha^i:\alpha\in B\}$ is linearly ordered, so it generates a branch through $S$ . This is a contradiction, and the proof is complete. ${\sf QED}$

Remark 6. One can easily modify the argument above to prove directly (without appealing to ${\sf MA}$ ) that ${\mathbb P}^\omega(S)$ is ccc. Hence, the only appeal to ${\sf MA}$ comes via arguing that there is already a splitting of $S$ into countably many antichains in the ground model.

Go to next talk.
Go to the intermezzo for a discussion of consistency strengths.

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