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 <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 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 <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): 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 <omega_2-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 <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_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 <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 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<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):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<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 alphain 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: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.


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