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 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 for a class of posets, stating that any admits filters meeting any given collection of many dense subsets of . The proper forcing axiom is of this kind, with being the class of proper posets. The strongest axiom to fall under this setting is Martin’s maximum , that has as the class of all posets preserving stationary subsets of .
Of particular interest is the `bounded’ version of these axioms, which, if posets in preserve , was shown by Bagaria to correspond precisely to an absoluteness statement, namely that for any .
In the next meeting I will review the notion of properness, and discuss some consequences of .
I want now to briefly sketch the following result from the Martin’s Maximum paper (Foreman, Magidor, Shelah. Martin’s Maximum, saturated ideals, and non-regular ultrafilters. Part I, Annals of Mathematics, 127 (1988), 1-47):
Say that a forcing preserves stationary subsets of (we will usually say that is stationary set preserving) iff there is some such that for any stationary set .
Theorem: Suppose that does not preserve stationary subsets of . Then fails.
Proof: Assume does not preserve subsets of . Begin by fixing -names and that are respectively interpreted in any -generic extension as a ground model-stationary subset of , and as a club subset of disjoint from .
Let . If is any filter and , then for some stationary set in the ground model. We now argue that if we could find such a meeting enough dense sets, then we could define (from ) a club subset (in the ground model) such that for some such and , which of course is a contradiction. We will ensure this by ensuring that if , then there is some which we may assume is below , and such that .
The set is easy to define: Simply take . We now proceed to exhibit many dense subsets of such that if were a filter meeting all of them (in addition to ), then indeed would be a club set. This proves that the forcing axiom for fails, as we want.
We are required to ensure that is unbounded in , and that it is closed. For let . Any filter meeting all the sets ensures that is unbounded.
But it also ensures that it is closed: Supposed otherwise. Then for some we have that . However, there is some such that and, for some , . However, there must be unboundedly many and for any of them there is some such that . It follows that and are incompatible, which contradicts that is a filter. This contradiction shows that is indeed closed, and completes the proof.