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)

Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216 89–117] and Steel [PFA implies AD L, J. Symbolic Logic70 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a singular strong (...) limit cardinal or PFA. (shrink)

We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that , the Axiom of Determinacy, holds in the of a generic extension of : every uncountable cardinal is singular, and every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let Ω be a regular uncountable cardinal. Let m < ω and M be an m-sound premouse and Σ be...

In this paper, we show that the failure of the unique branch hypothesis for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66 903–923; Core models with more Woodin cardinals, J. Symbolic Logic (...) 67 1197–1226] for tame trees. (shrink)

We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. The main result shows that these models are characterized by their correctness and the property that they correctly compute the tree from the appropriate scale. The main step towards this characterization shows that the tree from a scale can be reconstructed in a generic extension of an iterate of a mouse. We then construct models (...) with this property as generic extensions of iterates of mice under the assumption that the corresponding projective ordinal is below ω2ω2. On the way, we consider several related problems, including the question when forcing does not add equivalence classes to thin projective equivalence relations. For instance, we show that if every set has a sharp, then reasonable forcing does not add equivalence classes to thin provably View the MathML source Δ1/3 equivalence relations, and generalize this to all projective levels. (shrink)