## 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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