从 Reinhardt 基数得到带有 Woodin 基数的 Mice

Lightface mice with finitely many Woodin cardinals from optimal determinacy hypotheses

The determinacy of lightface $Δ^1_{2n+2}$ and boldface $\boldsymbolΠ^1_{2n+1}$ sets implies the existence of an $(ω, ω_1)$-iterable $M_{2n+1}^{\#}$.

Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

We prove the following result which is due to the third author. Let $n \geq 1$. If $\boldsymbolΠ^1_n$ determinacy and $Π^1_{n+1}$ determinacy both hold true and there is no $\boldsymbolΣ^1_{n+2}$-definable $ω_1$-sequence of pairwise distinct reals, then $M_n^\#$ exists and is $ω_1$-iterable. The proof yields that $\boldsymbolΠ^1_{n+1}$ determinacy implies that $M_n^\#(x)$ exists and is $ω_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \geq 1$, namely the statement that $\boldsymbolΠ^1_{n+1}$ determinacy implies $\Game^{(n)}(<ω^2 - \boldsymbolΠ…

Measures in Mice

This thesis analyses extenders in fine structural mice. Kunen showed that in the inner model for one measurable cardinal, there is a unique measure. This result is generalized, in various ways, to mice below a superstrong cardinal. The analysis is then used to show that certain tame mice satisfy $V=\mathsf{HOD}$. In particular, the approach proides a new proof of this result for the inner model $M_n$ for $n$ Woodin cardinals. It is also shown that in $M_n$, all homogeneously Suslin sets of reals are $\mathbfΔ^1_{n+1}$.

Determinacy and Large Cardinals

The study of inner models was initiated by Gödel's analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin's proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel's proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later N…

部分塔密封

Partial Tower Sealing

本文研究了部分塔密封的性质，证明其在特定模型中成立，并指出其一致性强度低于ZFC加上Woodin极限的存在。

Reinhardt基数与ZF的一致性

Inconsistency of Reinhardt cardinals with $\mathsf{ZF}$

本文证明了非平凡的Σ₁-初等嵌入j:V_{λ+3}→V_{λ+3}的存在性与ZF不一致。