## Set theory seminar -Forcing axioms and inner models III

(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

1.  Each $p_i$ is in $N$.
2. $p_{i+1}\le_i p_i$.
3. $p_i\in D_i$, 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 method of forcing ${\sf PFA}$ collapses 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.
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 heard that a student of Bagaria was thinking about this question, but I don’t know of any progress on it. 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 close. 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 qick 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 $(p_\xi,q_\xi)\in D_\xi=\{(p,\dot q):p\Vdash\dot q\in\tau_\xi\}$. The closure of ${\mathbb P}$ now allows us to pick a lower bound for this sequence, from which the proof is concluded as before.

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 $\omega_2$.

During the talk I did not finish the proof of this result, which I will continue in the next meeting. 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, with which I concluded this lecture:

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 for $\alpha<\omega_1$, $i,j. 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.