## 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 $Ssubseteqomega_1$ 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 $Nprec H_theta$ containing $dot X$, ${mathbb P}$, and any other relevant parameters. One can then produce a sequence $(p_n)_{ninomega}$ such that

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

Let $ple_i p_i$ for all $i$. Then $pVdash dot Xsubseteq N$, so $Ncap{sf ORD}$ 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 $-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, $-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 $-directed closed forcing.

Recall:

Definition. A poset ${mathbb P}$ is $-directed closed iff any directed $Din {mathcal P}_kappa({mathbb P})$ (i.e., for any $p,qin D$ there is $rin D$ with $rle p$ and $rle q$) admits a lower bound in ${mathbb P}$.

Clearly, any such forcing is $-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 $-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 for a sequence of $omega_1$ many dense subsets of $dot{mathbb Q}$.

For $xi, let $D_xi={(p,dot q): pVdashdot qintau_xi}$ and let $E_xi$ be the first-coordinate projection of $D_xi$.

By ${sf MM}$, there is a filter $Ksubseteq {mathbb P}*dot{mathbb Q}$ 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 $-directed closed, we can find a lower bound $p$ for $G$. Thus, $pVdashmbox{}{dot q:exists p'ge p,(p',q)incheck G}mbox{ is a filter meeting all the }tau_xi.mbox{''}$ ${sf QED}$

Remark 3. König and Yoshinobu have shown that for any $lambda$ there is a $-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 $p_0ge p_1gedots$ 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 $-closed poset ${mathbb P}$, a ${mathbb P}$-name $dot{mathbb Q}$ for a proper poset, and a ${mathbb P}$-name $(tau_alpha:alpha for a sequence of dense subsets of (the interpretation of) $dot{mathbb Q}$. As in Larson’s proof we can find a directed $Gsubseteq{mathbb P}$ 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 through $G$ such that for each $xi$ there is some $dot q_xi$ with $(p_xi,q_xi)in D_xi={(p,dot q):pVdashdot qintau_xi}$. 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 $j:Tto{mathbb R}$ 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 $D_t={(p_1,dots,p_k)in{mathbb P}^omega(S):tinbigcup_i p_i}$ for $tin S$, 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 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 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, $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 $alphain B$.

Consider $alpha, 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 ${r_alpha^i:alphain B}$ 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.