复活公理和令人振奋的红衣主教

Strongly uplifting cardinals and the boldface resurrection axioms

We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

Absoluteness via Resurrection

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velicković. We introduce a stronger form of resurrection axioms (the \emph{iterated} resurrection axioms $\textrm{RA}_α(Γ)$ for a class of forcings $Γ$ and a given ordinal $α$), and show that $\textrm{RA}_ω(Γ)$ implies generic absoluteness for the first-order theory of $H_{γ^+}$ with respect to forcings in $Γ$ preserving the axiom, where $γ=γ_Γ$ is a cardinal which depends on $Γ$ ($γ_Γ=ω_1$ if $Γ$ is any among the classes of countably closed, proper, semiproper, stationar…

Weakly extendible cardinals and compactness of extended logics

We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V=L, a cardinal $κ$ is the weak compactness number of ${\cal L}^{\aleph_0,II}_{stat,κ,ω}$ if and only if it is th…

Laver-一般大基数下的极大性原理与复活公理

Maximality Principles and Ressurection Axioms under a Laver-generic large cardinal

本文研究了 Laver-一般性公理与极大性原理及复活公理的关系，证明无参数和参数化极大性原理独立于 Laver-一般性公理，而局部极大性原理和复活公理可由足够强的 tightly Laver-一般大基数存在性推出。

可扩展基数，以及用于可扩展性的 Laver-生成大基数公理

Extendible cardinals, and Laver-generic large cardinal axioms for extendibility

引入用于可扩展性的Laver-生成大基数公理，证明其能推导出先前已知的大多数推论，并探讨了其一致性、反射性质及相关分离结果。