We re-prove Becker’s theorem from Becker :229–234, 1981) by showing that \}\) implies that \\vDash ``\omega _2\) is -supercompact”. Our proof uses inner model theoretic tools instead of Baire category. We also show that \ is \-strongly compact.

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf (...) over}$ - ${\sf uB}$ be the statement that in all generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$. In fact, we isolate an exact theory that is equiconsistent with both. As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$, called $\sf {Tower \ Sealing}$, is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$. (shrink)

Assuming $V=L+AD$, using methods from inner model theory, we give a new proof of the strong partition property for ${\sim}{ \delta }^{2}_{1}$. The result was originally proved by Kechris et al.

Working in the theory “ZF + There is a nontrivial elementary embedding j: V → V ”, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ (...) ≥ ℵ2 satisfies the square bracket infinite exponent partition relation μ → [μ]ωℵ2. We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j : V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice. (shrink)

Volume 19, Issue 9, September 2019, Page W1-W2.

The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture. One particular motivation for resolving MSC is that (...) it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large. (shrink)