580 -Cardinal arithmetic (4)

February 11, 2009

2. Silver’s theorem.

From the results of the previous lectures, we know that any power \kappa^\lambda can be computed from the cofinality and gimel functions (see the Remark at the end of lecture II.2). What we can say about the numbers \gimel(\lambda) varies greatly depending on whether \lambda is regular or not. If \lambda is regular, then \gimel(\lambda)=2^\lambda. As mentioned on lecture II.2, forcing provides us with a great deal of freedom to manipulate the exponential function \kappa\mapsto 2^\kappa, at  least for \kappa regular. In fact, the following holds:

Theorem 1. (Easton). If {\sf GCH} holds, then for any definable function F from the class of infinite cardinals to itself such that:

  1. F(\kappa)\le F(\lambda) whenever \kappa\le\lambda, and
  2. \kappa<{\rm cf}(F(\kappa)) for all \kappa,

there is a class forcing {\mathbb P} that preserves cofinalities and such that in the extension by {\mathbb P} it holds that 2^\kappa=F^V(\kappa) for all regular cardinals \kappa; here, F^V is the function F as computed prior to the forcing extension. \Box

For example, it is consistent that 2^\kappa=\kappa^{++} for all regular cardinals \kappa (as mentioned last lecture, the same result is consistent for all cardinals, as shown by Foreman and Woodin, although their argument is significantly more elaborate that Easton’s). There is almost no limit to the combinations that the theorem allows: We could have 2^\kappa=\kappa^{+16} whenever \kappa=\aleph_\tau is regular and \tau is an even ordinal, and 2^\kappa=\kappa^{+17} whenever \kappa=aleph_\tau for some odd ordinal \tau. Or, if there is a proper class of weakly inaccessible cardinals (regular cardinals \kappa such that \kappa=\aleph_\kappa) then we could have 2^\kappa= the third weakly inaccessible strictly larger than \kappa, for all regular cardinals \kappa, etc.

Morally, Easton’s theorem says that there is nothing else to say about the gimel function on regular cardinals, and all that is left to be explored is the behavior of \gimel(\lambda) for singular \lambda. In this section we begin this exploration. However, it is perhaps sobering to point out that there are several weaknesses in Easton’s result.

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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.

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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.

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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.

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Set theory seminar -Forcing axioms and inner models III

October 1, 2008

(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.

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Set theory seminar -Forcing axioms and inner models -Intermezzo

September 30, 2008

This posting complements a series of talks given at the Set Theory Seminar at BSU from September 12 to October 24, 2008. Here is a list of links to the talks in this series:

  • First talk, September 12, 2008.
  • Second talk, September 19, 2008.
  • Third talk, September 26, 2008.
  • Fourth talk, October 3, 2008.
  • Fifth talk, October 10, 2008.
  • Sixth talk, October 17, 2008.
  • Seventh talk, October 24, 2008.

[Version of October 31.]

I’ll use this post to provide some notes about consistency strength of the different natural hierarchies that forcing axioms and their bounded versions suggest. This entry will be updated with some frequency until I more or less feel I don’t have more to add. Feel free to email me additions, suggestions and corrections, or to post them in the comments. In fact, please do.

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Set theory seminar -Forcing axioms and inner models

September 12, 2008

Today I started a series of talks on “Forcing axioms and inner models” in the Set Theory Seminar. The goal is to discuss a few results about strong forcing axioms and to see how these axioms impose a certain kind of `rigidity’ to the universe.

I motivated forcing axioms as trying to capture the intuition that the universe is `wide’ or `saturated’ in some sense, the next natural step after the formalization via large cardinal axioms of the intuition that the universe is `tall.’

The extensions of {\sf ZFC} obtained via large cardinals and those obtained via forcing axioms share a few common features that seem to indicate their adoption is not arbitrary. They provide us with reflection principles (typically, at the level of the large cardinals themselves, or at small cardinals, respectively), with regularity properties (and determinacy) for many pointclasses of reals, and with generic absoluteness principles.

The specific format I’m concentrating on is of axioms of the form {\sf FA}({\mathcal K}) for a class {\mathcal K} of posets, stating that any {\mathbb P}\in{\mathcal K} admits filters meeting any given collection of \omega_1 many dense subsets of {\mathbb P}. The proper forcing axiom {\sf PFA} is of this kind, with {\mathcal K} being the class of proper posets. The strongest axiom to fall under this setting is Martin’s maximum {\sf MM}, that has as {\mathcal K} the class of all posets preserving stationary subsets of \omega_1.

Of particular interest is the `bounded’ version of these axioms, which, if posets in {\mathcal K} preserve \omega_1, was shown by Bagaria to correspond precisely to an absoluteness statement, namely that H_{\omega_2}\prec_{\Sigma_1}V^{\mathbb P} for any {\mathbb P}\in{\mathcal K}.

In the next meeting I will review the notion of properness, and discuss some consequences of {\sf BPFA}.

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