Set theory seminar – Marion Scheepers: Coding strategies (IV)

October 12, 2010

For the third talk (and a link to the second one), see here. The fourth talk took place on October 12.

We want to show the following version of Theorem 2:

Theorem. Suppose \kappa is a singular strong limit cardinal of uncountable cofinality. Then the following are equivalent:

  1. For each ideal J on \kappa, player II has a winning coding strategy in RG(J).
  2. 2^\kappa<\kappa^{+\omega}.

Since 2^\kappa has uncountable cofinality, option 2 above is equivalent to saying that the instance of {\sf wSCH} corresponding to \kappa holds.

Before we begin the proof, we need to single out some elementary consequences in cardinal arithmetic of the assumptions on \kappa. First of all, since \kappa is singular strong limit, then for any cardinal \lambda<\kappa, we have that

\kappa^\lambda=\left\{\begin{array}{cl}\kappa&\mbox{\ if }\lambda<{\rm cf}(\kappa),\\ 2^\kappa&\mbox{\ otherwise.}\end{array}\right.

Also, since the cofinality of \kappa is uncountable, we have Hausdoff’s result that if n<\omega, then (\kappa^{+n})^{\aleph_0}=\kappa^{+n}. I have addressed both these computations in my lecture notes for Topics in Set Theory, see here and here.

We are ready to address the Theorem.

Proof. (2.\Rightarrow 1.) We use Theorem 1. If option 1. fails, then there is an ideal J on \kappa with {\rm cf}(\left< J\right>,{\subset})>|J|.

Note that {\rm cf}(\left< J\right>,{\subset})\le({\rm cf}(J,{\subset}))^{\aleph_0}, and \kappa\le|J|. Moreover, if \lambda<\kappa, then 2^\lambda<{\rm cf}(J,{\subset}) since, otherwise,

({\rm cf}(J,{\subset}))^{\aleph_0}\le 2^{\lambda\aleph_0}=2^\lambda<\kappa.

So {\rm cf}(J,{\subset})\ge\kappa and then, by Hausdorff, in fact {\rm cf}(J,{\subset})\ge \kappa^{+\omega}, and option 2. fails.

(1.\Rightarrow 2.) Suppose option 2. fails and let \lambda=\kappa^{+\omega}, so \kappa<\lambda<2^\kappa and {\rm cf}(\lambda)=\omega. We use \lambda to build an ideal J on \kappa with {\rm cf}(\left< J\right>,{\subset})>|J|.

For this, we use that there is a large almost disjoint family of functions from {\rm cf}(\kappa) into \kappa. Specifically:

Lemma. If \kappa is singular strong limit, there is a family {\mathcal F}\subseteq{}^{{\rm cf}(\kappa)}\kappa with {}|{\mathcal F}|=2^\kappa and such that for all distinct f,g\in{\mathcal F}, we have that {}|\{\alpha<{\rm cf}(\kappa)\mid f(\alpha)=g(\alpha)|<{\rm cf}(\kappa).

In my notes, I have a proof of a general version of this result, due to Galvin and Hajnal, see Lemma 12 here; essentially, we list all functions f:{\rm cf}(\kappa)\to\kappa, and then replace them with (appropriate codes for) the branches they determine through the tree \kappa^{{\rm cf}(\kappa)}. Distinct branches eventually diverge, and this translates into the corresponding functions being almost disjoint.

Pick a family {\mathcal F} as in the lemma, and let {\mathcal G} be a subfamily of size \lambda. Let S=\bigcup{\mathcal G}\subseteq{\rm cf}(\kappa)\times\kappa. We proceed to show that |S|=\kappa and use {\mathcal G} to define an ideal J on S as required.

First, obviously |S|\le\kappa. Since \kappa<\lambda=|{\mathcal G}| and {\mathcal G}\subseteq{\mathcal P}(S), it follows that {}|S|\ge\kappa, or else {}|{\mathcal P}(S)|<\kappa, since \kappa is strong limit.

Now define

J=\{X\subseteq S\mid\exists {\mathcal H}\subseteq{\mathcal G}\,(|{\mathcal H}|<\omega,\bigcup{\mathcal H}\supseteq X)\}.

Clearly, J is an ideal. We claim that |J|=\lambda. First, each singleton \{f\} with f\in{\mathcal G} is in J, so {}|J|\ge\lambda. Define \Phi:[{\mathcal G}]^{<\aleph_0}\to J by \Phi({\mathcal H})=\bigcup{\mathcal H}). Since the functions in {\mathcal G} are almost disjoint, it follows that \Phi is 1-1. Let G be the image of \Phi. By construction, G is cofinal in J. But then

{}|J|\le|{\mathcal G}|2^{{\rm cf}(\kappa)}=\lambda 2^{{\rm cf}(\kappa)}=\lambda,

where the first inequality follows from noticing that any X\in J has size at most {\rm cf}(\kappa). It follows that |J|=\lambda, as claimed.

Finally, we argue that {\rm cf}(\left< J\right>,{\subset})>\lambda, which completes the proof. For this, consider a cofinal {\mathcal A}\subseteq\left< J\right>, and a map f:{\mathcal A}\to[{\mathcal G}]^{\le\aleph_0} such that for all A\in{\mathcal A}, we have A\subseteq\bigcup f(A).

Since {\mathcal A} is cofinal in \left< J\right>, it follows that f[{\mathcal A}] is cofinal in {}[{\mathcal G}]^{\le\aleph_0}. But this gives the result, because

{}|{\mathcal A}|\ge{\rm cf}([{\mathcal G}]^{\le \aleph_0},{\subset})={\rm cf}([\lambda]^{\le \aleph_0},{\subset})>\lambda,

and we are done. \Box


Set theory seminar – Marion Scheepers: Coding strategies (III)

September 28, 2010

For the second talk (and a link to the first one), see here. The third talk took place on September 28.

In the second case, we fix an X\in J with {}|J(X)|<|J|. We can clearly assume that S is infinite, and it easily follows that {}|{\mathcal P}(X)|=|J|. This is because any Y\in J can be coded by the pair (Y\cap X,X\cup(Y\setminus X)), and there are only {}|J(X)| many possible values for the second coordinate.

In particular, X is infinite, and we can fix a partition X=\bigsqcup_n X _n of X into countably many pieces, each of size {}|X|. Recall that we are assuming that \text{cf}(\left< J\right>,{\subset})\le|J| and have fixed a set H cofinal in \left< J\right> of smallest possible size. We have also fixed a perfect information winning strategy \Psi for II, and an f:\left< J\right>\to H with A\subseteq f(A) for all A.

For each n, fix a surjection f:{\mathcal P}(X_n)\setminus\{\emptyset,X_n\}\to{}^{<\omega}H.

We define F:J\times\left< J\right>\to J as follows:

  1. Given O\in \left< J\right>, let

    A=\Psi(\left<f(O)\right>)\setminus X,


    B\in{\mathcal P}(X_0)\setminus\{\emptyset, X_0\} such that f_0(B)=\left< f(O)\right>,

    and set F(\emptyset,O)=A\cup B.

  2. Suppose now that (T,O)\in J\times\left< J\right>, that T\ne\emptyset , and that there is an n such that \bigcup_{j<n} X_j\subseteq T, T\cap X_n\ne\emptyset,X_n, and T\cap\bigcup_{k>n}X_k=\emptyset. Let

    B\in{\mathcal P}(X_{n+1})\setminus\{\emptyset,X_{n+1}\} be such that f_{n+1}(B)=f_n(T\cap X_n){}^\frown\left< f(O)\right>,


    A=\Psi(f_{n+1}(B))\setminus X

    and set F(T,O)=A\cup\bigcup_{j\le n}X_j\cup B.

  3. Define F(T,O)=\emptyset in other cases.

A straightforward induction shows that F is winning. The point is that in a run of the game where player II follows F:

  • Player II’s moves code the part that lies outside of X of player II’s moves in  a run {\mathcal A} of the game following \Psi where I plays sets covering the sets in the original run. For this, note that at any inning there is a unique index n such that player II’s move covers \bigcup_{j<n}X_j, is disjoint from \bigcup_{j>n}X_j, and meets X_n in a set that is neither empty nor all of X_n, and this n codes the inning of the game, and the piece of player II’s move in X_n codes the history of the run {\mathcal A} played so far.
  • X is eventually covered completely, so in particular the parts inside X of player II’s responses in the run {\mathcal A} are covered as well.

This completes the proof of Theorem 1. \Box

By way of illustration, consider the case where J is the ideal of finite sets of some set S. Then whether II has a winning coding strategy turns into the question of when it is that \text{cf}({\mathcal P}_{\aleph_1}(S))\le|S|. This certainly holds if {}|S|={\mathfrak c} or if {}|S|<\aleph_\omega. However, it fails if {}|S|=\aleph_\omega.

This example illustrates how player II really obtains an additional advantage when playing in WMG(J) rather than just in RG(J). To see that this is the case, consider the same J as above with {}|S|=\aleph_\omega. This is an instance of the countable-finite game. We claim that II has a winning coding strategy in this case. To see this, consider a partition of S into countably many sets S_n with {}|S_n|=\aleph_n. For each n, pick a winning coding strategy \sigma_n for the countable-finite game on S_n, and define a strategy in WMG(J) so that for each n it simulates a run of the game WMG(J\cap{\mathcal P}(S_n) with II following \sigma_n, as follows: In inning n, II plays on S_i for i\le n; player I’s moves in the “i-th board” are the intersection with S_i of I’s moves in WMG(J), and I’s first move occurred at inning i. (II can keep track of n in several ways, for example, noticing that, following the proof of Theorem 1 produces coding strategies that never play the empty set.)

Note that this strategy is not winning in RG(J), the difference being that there is no guarantee that (for any i) the first i moves of I in the (i+1)-st board are going to be covered by II’s responses. On the other hand, the strategy is winning in WMG(J), since, no matter how late one starts to play on the i-th board, player I’s first move covers I’s prior moves there (and so, II having a winning coding strategy for the game that starts with this move, will also cover those prior moves).

The first place where this argument cannot be continues is when |S|=\aleph_{\omega_1}. However, {\sf GCH} suffices to see that player I has a winning strategy in RG(J) in this case, and so we can continue. This illustrates the corollary stated in the first talk, that {\sf GCH} suffices to guarantee that II always has a winning coding strategy in WMG(J).

The natural question is therefore how much one can weaken the {\sf GCH} assumption, and trying to address it leads to Theorem 2, which will be the subject of the next (and last) talk.

Set theory seminar – Marion Scheepers: Coding strategies (II)

September 27, 2010

For the first talk, see here. The second talk took place on September 21.

We want to prove (2.\Rightarrow1.) of Theorem 1, that if {\rm cf}(\left< J\right>,\subset)\le|J|, then II has a winning coding strategy in RG(J).

The argument makes essential use of the following:

Coding Lemma. Let ({\mathbb P},<) be a poset such that for all p\in{\mathbb P},

{}|\{q\in{\mathbb P}\mid q>p\}|=|{\mathbb P}|.

Suppose that {}|H|\le|{\mathbb P}|. Then there is a map \Phi:{\mathbb P}\to{}^{<\omega}H such that

\forall p\in{\mathbb P}\,\forall\sigma\in H\,\exists q\in{\mathbb P}\,(q>p\mbox{ and }\Phi(q)=\sigma).

Proof. Note that {\mathbb P} is infinite. We may then identify it with some infinite cardinal \kappa. It suffices to show that for any partial ordering \prec on \kappa as in the hypothesis, there is a map \Phi:\kappa\to\kappa such that for any \alpha,\beta, there is a \gamma with \alpha\prec\gamma such that \Phi(\gamma)=\beta.

Well-order \kappa\times\kappa in type \kappa, and call R this ordering. We define \Phi by transfinite recursion through R. Given (\alpha,\beta), let A be the set of its R-predecessors,

A=\{(\mu,\rho)\mid(\mu,\rho) R(\alpha,\beta)\}.

Our inductive assumption is that for any pair (\mu,\rho)\in A, we have chosen some \tau with \mu\prec\tau, and defined \Phi(\tau)=\rho.  Let us denote by D_A the domain of the partial function we have defined so far. Note that {}|D_A|<\kappa. Since \{\gamma\mid\alpha\prec\gamma\} has size \kappa, it must meet \kappa\setminus D_A. Take \mu to be least in this intersection, and set \Phi(\mu)=\beta, thus completing the stage (\alpha,\beta) of this recursion.

At the end, the resulting map can be extended to a map \Phi with domain \kappa in an arbitrary way, and this function clearly is as required. \Box

Back to the proof of (2.\Rightarrow1.). Fix a perfect information winning strategy \Psi for II in RG(J), and a set H cofinal in \left< J\right> of least possible size. Pick a f:\left< J \right>\to H such that for all A\in \left< J\right> we have A\subseteq f(A).

Given X\in J, let J(X)=\{Y\in J\mid X\subseteq Y\}. Now we consider two cases, depending on whether for some X we have {}|J(X)|<|J| or not.

Suppose first that |J(X)|=|J| for all X. Then the Coding Lemma applies with (J,\subset) in the role of {\mathbb P}, and H as chosen. Let \Phi be as in the lemma.

We define F:J\times\left< J\right>\to J as follows:

  1. Given O\in\left< J\right>, let Y\supseteq\psi(f(O)) be such that \Phi(Y)=\left<f(O\right>, and set F(\emptyset,O)=Y.
  2. Given (T,O)\in J\times\left< J\right> with T\ne\emptyset, let Y\supseteq \Psi(\Phi(T){}^\frown\left<f(O)\right>) be such that \Phi(Y)=\Phi(T){}^\frown\left< f(O)\right>, and set F(T,O)=Y.

Clearly, F is winning: In any run of the game with II following F, player II’s moves cover their responses following \Psi, and we are done since \Psi is winning.

The second case, when there is some X\in J with |J(X)|<|J|, will be dealt with in the next talk.

Set theory seminar – Marion Scheepers: Coding strategies (I)

September 25, 2010

This semester, the seminar started with a series of talks by Marion. The first talk happened on September 14.

We consider two games relative to a (proper) ideal J\subset{\mathcal P}(S) for some set S. The ideal J is not assumed to be \sigma-complete; we denote by \left< J\right> its \sigma-closure, i.e., the collection of countable unions of elements of J. Note that \left< J\right> is a \sigma-ideal iff \left< J\right> is an ideal iff S\notin\left< J\right>.

The two games we concentrate on are the Random Game on J, RG(J), and the Weakly Monotonic game on J, WMG(J).

In both games, players I and II alternate for \omega many innings, with I moving first, moving as follows:

\begin{array}{cccccc} I&O_0\in\left< J\right>&&O_1\in\left< J_2\right>&&\cdots\\ II&&T_0\in J&&T_1\in J \end{array}

In RG(J) we do not require that the O_i relate to one another in any particular manner (thus “random”), while in WMG(J) we require that O_1\subseteq O_2\subseteq\dots (thus “weakly”, since we allow equality to occur).

In both games, player II wins iff \bigcup_n T_n\supseteq\bigcup_n O_n. Obviously, II has a (perfect information) winning strategy, with = rather than the weaker \supseteq.

However, we are interested in an apparently very restrictive kind of strategy, and so we will give some leeway to player II by allowing its moves to over-spill if needed. The strategies for II we want to consider we call coding strategies. In these strategies, II only has access to player I’s latest move, and to its own most recent move. So, if F is a coding strategy, and II follows it in a run of the game, then we have that for every n,


with T_{-1}=\emptyset.

The underlying goal is to understand under which circumstances player II has a winning coding strategy in WMG(J). Obviously, this is the case if II has a winning coding strategy in RG(J).

Theorem 1. For an ideal J\subset{\mathcal P}(S), the following are equivalent:

  1. II has a winning coding strategy in RG(J).
  2. {\rm cf}(\left< J\right>,{\subset})\le|J|.

Corollary. {\sf GCH} implies that for any ideal J\subset{\mathcal P}(S), II has a winning strategy in WMG(J).

We can reformulate our goal as asking how much one can weaken {\sf GCH} in the corollary.

Let’s denote by {\sf wSCH}, the weak singular cardinals hypothesis, the statement that if \kappa is singular strong limit of uncountable cofinality, then for no cardinal \lambda of countable cofinality, we have \kappa<\lambda<2^\kappa.

By work of Gitik and Mitchell, we know that the negation of {\sf wSCH} is equiconsistent with the existence of a \kappa of Mitchell order o(\kappa)=\kappa^{+\omega}+\omega_1.

Theorem 2. The following are equivalent:

  1. {\sf wSCH}.
  2. For each ideal J on a singular strong limit \kappa of uncountable cofinality, II has a winning strategy in RG(J).

We now begin the proof of Theorem 1.

(1.\Rightarrow2.) Suppose II has a winning coding strategy F in RG(J). We want to show that {\rm cf}(\left< J\right>,{\subset})\le|J|. For this, we will define a map f:J\to\left< J\right> with \subset-cofinal range, as follows: Given X\in J, let T_0=X and T_{n+1}=F(T_n,\emptyset) for all n. Now set

f(X)=\bigcup_n T_n.

To see that f is cofinal, given O\in\left< J\right>, let X=F(\emptyset,O), so that the T_n are II’s responses using F in a run of the game where player I first plays O and then plays \emptyset in all its subsequent moves. Since F is winning, we must have f(X)\supseteq O.

Next talk.

Set theory seminar -Richard Ketchersid: Quasiiterations I. Iteration trees

January 19, 2009

In October 24-November 14 of 2008, Richard Ketchersid gave a nice series of talks on quasiiterations at the Set Theory Seminar. The theme is to correctly identify `nice’ branches through iteration trees, and to see how difficult it is for a model to compute these branches. Richard presented a prototypical result in this area (due to Woodin) and a nice application (due to Jackson and Ketchersid). This post will be far from self-contained, and only present some of the definitions.

[Edit Sep. 25, 2010: My original intention was to follow this post with two more notes, on Woodin’s result and on the Jackson-Ketchersid theorem, but I never found the time to polish the presentation to a satisfactory level, so instead I will let the interested reader find my drafts at Lucien’s library.]

I’ll assume known the notions of extender and Woodin cardinal, and associated notions like the length or strength of an extender. A good reference for this post is Donald Martin, John Steel, Iteration trees, Journal of the American Mathematical Society 7 (1) 1994, 1-73. As usual, all inaccuracies below are mine. Some of the notions below are slightly simpler than the official definitions. These notions are all due to Donald Martin, John Steel, and Hugh Woodin.

In this post I present the main notions (iteration trees and iterability) and close with a quick result about the height of tree orders. The order I follow is close to Richard’s but it differs from his presentation at a few places.

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Set theory seminar -Stefan Geschke: Cofinalities of algebraic structures

January 6, 2009

This is a short overview of a talk given by Stefan Geschke on November 21, 2008. Stefan’s topic, Cofinalities of algebraic structures and coinitialities of topological spaces, very quickly connects set theory with other areas, and leads to well-known open problems. In what follows, compact always includes Hausdorff. Most of the arguments I show below are really only quick sketches rather than complete proofs. Any mistakes or inaccuracies are of course my doing rather than Stefan’s, and I would be grateful for comments, corrections, etc.

Definition. Let A be a (first order) structure in a countable language. Write {\rm cf}(A) for the smallest \delta such that A=\bigcup_{\alpha<\delta}A_\alpha for a strictly increasing union of proper substructures.

Since the structures A_\alpha need to be proper, {\rm cf}(A) is not defined if A is finite. It may also fail to exist if A is countable, but it is defined if A is uncountable. Moreover, if {\rm cf}(A) exists, then

  1. {\rm cf}(A)\le|A|, and
  2. {\rm cf}(A) is a regular cardinal.

Example 1. Groups can have arbitrarily large cofinality. This is not entirely trivial, as the sets A_\alpha may have size |A|.

Question 1. Is every regular cardinal realized this way?

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Set theory seminar -Forcing axioms and inner models VII

October 24, 2008

I concluded my series of talks by showing the following theorem of Viale:

Theorem (Viale). Assume {\sf CP}(\kappa^+) and let M\subseteq V be an inner model where \kappa is regular and such that (\kappa^+)^M=\kappa^+. Then {\rm cf}(\kappa)\ne\omega.

This allows us to conclude, via the results shown last time, that if {\sf PFA} holds in V and M computes cardinals correctly, then it also computes correctly ordinals of cofinality \omega.

An elaboration of this argument is expected to show that, at least  if we strengthen the assumption of {\sf PFA} to {\sf MM}, then M computes correctly ordinals of cofinality \omega_1.

Under an additional assumption, Viale has shown this:  If {\sf MM} holds in V, \kappa is a strong limit cardinal, (\kappa^+)^M=\kappa^+, and in M we have that \kappa is regular, then in V the cofinality of \kappa cannot be \omega_1. The new assumption on \kappa allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and I_{<f} [\lambda], Fund. Math. 148 (1995), 165-198, in place of the structure imposed by {\sf CP}(\kappa^+). It is still open if the corresponding covering statement {\sf CP}(\kappa^+,\omega_1) follows from {\sf MM}, which would eliminate the need for this the strong limit requirement.

  • Go to the intermezzo for a discussion of consistency strengths.

Set theory seminar -Forcing axioms and inner models VI

October 17, 2008

I presented a sketch of a nice proof due to Todorcevic that {\sf PFA} implies the P-ideal dichotomy {\sf PID}. I then introduced Viale’s covering property {\sf CP} and showed that it follows from {\sf PID}. Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:

Theorem (Viale). Assume M\subseteq V is an inner model.

  1. If {\sf PFA} holds in V and M computes cardinals correctly, then it also computes correctly ordinals of cofinality \omega.
  2. If {\sf MM} holds in V, \kappa is a strong limit cardinal, (\kappa^+)^M=\kappa^+, and in M we have that \kappa is regular, then in V, the cofinality of \kappa cannot be \omega_1.

It follows from this result and the last theorem from last time that if V is a model of {\sf MM} and a forcing extension of an inner model M by a cardinal preserving forcing, then {\sf ORD}^{\omega_1}\subset M.

In fact, the argument from last time shows that we can weaken the assumption that V is a forcing extension to the assumption that for all \kappa there is a regular cardinal \lambda\ge\kappa such that  in M we have a partition S^\lambda_\omega=\sqcup_{\alpha<\kappa}S_\alpha where each S_\alpha is stationary in V.

It is possible that this assumption actually follows from {\sf MM} in V. However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable V cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of V that preserves cardinals, does not add reals, and (for some cardinal \kappa) the set of points of countable V-cofinality in \lambda is nonstationary for every regular \lambda\ge\kappa^+. Obviously, this situation is incompatible with {\sf PFA} in V, by Viale’s result.

Set theory seminar -Forcing axioms and inner models V

October 12, 2008

We showed Velickovic’s result that under {\sf MM} any inner model that computes \omega_2 correctly actually contains H_{\omega_2}.

The argument depends on the (weak) reflection principle (a consequence of {\sf MM}) and a combinatorial result due to Gitik.

It is open whether this result holds with {\sf PFA} in place of {\sf MM}, but an attempt to settle this led to the discovery that {\sf BPFA} implies the existence of a definable (in a subset of \omega_1) well-ordering of the reals. The well-ordering is actually \Delta_1 in the parameter, and the proof shows that H_{\omega_2} can be decomposed as a union of small transitive structures whose height determines their reals. This “L-like” decomposition of H_{\omega_2} is expected to continue for larger cardinals, which leads to the following:

Conjecture (Caicedo, Velickovic). Assume {\sf MM} and let M be an inner model that computes cardinals correctly. Then {\sf ORD}^{\omega_1}\subset M.

Although the conjecture is still open, there is (significant) partial evidence suggesting it. For example, we showed that if V satisfies {\sf MM} and is a forcing extension of an inner model that computes correctly the class of ordinals of cofinality \omega_1, then {\sf ORD}^{\omega_1}\subset M.

Set theory seminar -Forcing axioms and inner models IV

October 3, 2008

We proved Baumgartner’s result that under {\sf BPFA}, every tree of height and size \omega_1 is sealed in the sense that no outer model can add a new uncountable branch. From this we concluded Todorcevic’s result that under {\sf BPFA} any forcing adding a subset of \omega_1 either adds a real or else it collapses \omega_2. We also drew some conclusions about inner models of {\sf GCH}.