An inner model proof of the strong partition property of \delta^2_1

We prove the strong partition property for \delta^2_1 using methods from inner model theory.Comment: submitte

Negative results on precipitous ideals on omega_1

We show that in extender models there are no generic embeddings with critical point $\omega_1$ that resemble the stationary tower at the second Woodin cardinal

The Largest Suslin Axiom

We develop the basic fine structure theory of the minimal model of the Largest Suslin Axiom. In particular, we prove that that the minimal model of the Largest Suslin Axiom satisfies the Mouse Set Conjecture, and that the Proper Forcing Axiom implies the minimal model of the Largest Suslin Axiom exists

Unreachability of Inductive-Like Pointclasses in L(R)L(\mathbb{R})

Hjorth proved from $ZF + AD + DC$ that there is no sequence of distinct $\Sigma^1_2$ sets of length $\delta^1_2$. Sargsyan extended Hjorth's technique to show there is no sequence of distinct $\Sigma^1_{2n}$ sets of length $\delta^1_{2n}$. Sargsyan conjectured an analogous property is true for any regular Suslin pointclass in $L(R)$ -- i.e. if $\kappa$ is a regular Suslin cardinal in $L(R)$, then there is no sequence of distinct $\kappa$-Suslin sets of length $\kappa^+$ in $L(R)$. We prove this in the case that the pointclass $S(\kappa)$ is inductive-like

Sealing of the universally Baire sets

This article outlines the proof that over some mild large cardinal theory, Sealing is equiconsistent with Largest Suslin Axiom (LSA)-over-uB. Let LSA-over-uB be the statement that in all (set) generic extensions there is a model of LSA whose Suslin, co-Suslin sets are the universally Baire sets