\nabla_κ, remarkable cardinals, and 0^#

Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle

We consider a weak version of Schindler's remarkable cardinals that may fail to be $Σ_2$-reflecting. We show that the $Σ_2$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and we show that the existence of a non-$Σ_2$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an $ω$-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + "Ord is not $Δ_2$-Mahlo" and $\text{gVP}({\bfΠ}_1)$ + "there is no proper clas…

Indestructibility properties of remarkable cardinals

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the indestructibility properties of remarkable cardinals. We show that if $κ$ is remarkable, then there is a forcing extension in which the remarkability of $κ$ becomes indestructible by all $\ltκ$-closed $\leqκ$-distributive forcing and all two-step iterations of the form ${\rm Add}(κ,θ)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\ltκ$-closed and $\leqκ$-di…

L(R) absoluteness under proper forcings

We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness under proper forcings. In particular, said absoluteness does not imply Pi^1_1 determinacy.

力迫构造$Z_3$ + Harrington原理的集合模型

Force a set model of $Z_3$ + Harrington's Principle

本文在存在remarkable基数且其上有弱不可达基数的假设下，通过无需重塑的集合力迫方法，构造了满足三阶算术（$Z_3$）加Harrington原理（HP）的集合模型。

Martin 的可测膨胀子

Martin's measurable dilator

本文从 Martin 利用可迭代秩到秩嵌入证明 $\mathbfΠ^1_2$-决定性的论证中，提取并界定可测膨胀子这一大基数对象，以揭示其证明的结构本质。

The Ostaszewski square, and homogenous Souslin trees

Assume GCH and let $λ$ denote an uncountable cardinal. We prove that if $\square_λ$ holds, then this may be witnessed by a coherent sequence $< C_α| α< λ^+ >$ with the following remarkable guessing property: For every sequence $< A_i | i<λ>$ of unbounded subsets of $λ^+$, and every limit $θ<λ$, there exists some $α<λ^+$ such that $\otp(C_α)=θ$, and the $(i+1)_{th}$-element of $C_α$ is a member of $A_i$, for all $i<θ$. As an application, we construct an homogenous $λ^+$-Souslin tree from $GCH+\square_λ$, for every singular cardinal $λ$. In addition, as a by-product, a theorem of Farah and Velic…