Set theory seminar -Forcing axioms and inner models V

• Go to previous talk.

We showed Velickovic’s result that under any inner model that computes correctly actually contains .

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

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

Conjecture (Caicedo, Velickovic). Assume and let be an inner model that computes cardinals correctly. Then .

Although the conjecture is still open, there is (significant) partial evidence suggesting it. For example, we showed that if satisfies and is a forcing extension of an inner model that computes correctly the class of ordinals of cofinality , then .

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

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