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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