Set theory seminar -Forcing axioms and inner models V

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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3 Responses to Set theory seminar -Forcing axioms and inner models V

  1. […] follows from this result and the last theorem from last time that if is a model of and a forcing extension of an inner model by a cardinal preserving […]

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