$\mathsf {Sealing}$ from iterability

Abstract:

We show that if $V$ has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then $\mathsf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that $\mathsf {Sealing}$ holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that $\mathsf {Sealing}$ is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of $\mathsf {Sealing}$ obtained by W.H. Woodin.

The $\mathsf {Largest\ Suslin\ Axiom}$ ($\mathsf {LSA}$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\mathsf {LSA\text {-}over\text {-}uB}$ be the statement that in all (set) generic extensions there is a model of $\mathsf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming $V$ has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe $V[g]$, where $g$ is $V$-generic for the collapse of the successor of the least strong cardinal to be countable, the theory $\mathsf {LSA}$-$\mathsf {over}$-$\mathsf {UB}$ fails; this implies that $\mathsf {LSA\text {-}over\text {-}UB}$ is not equivalent to $\mathsf {Sealing}$ (over the base theory of $V[g]$). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “$\mathsf {AD}^{L(\mathbb {R})}$ holds in all generic extensions” is equivalent to “the theory of $L(\mathbb {R})$ is sealed” in the presence of a proper class of measurable cardinals.