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.


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<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.

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