Indestructibility properties of remarkable cardinals

Abstract

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 $\kappa$ is remarkable, then there is a forcing extension in which the remarkability of $\kappa$ becomes indestructible by all $<\kappa$-closed $\le \kappa$-distributive forcing and all two-step iterations of the form $\mathrm{Add}(\kappa ,\theta )\ast \dot{\mathbb{R}}$, where $\dot{\mathbb{R}}$ is forced to be $<\kappa$-closed and $\le \kappa$-distributive. In the process, we introduce the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. We also show that remarkability is preserved by the canonical forcing of the $\mathrm{GCH}$.

21 pages. Commentary concerning this article can be made at http://boolesrings.org/victoriagitman/2014/09/17/indestructible-remarkable-cardinals/

References in corpus (1)

• Woodin's surgery method