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 $\lt\kappa$-closed $\leq\kappa$-distributive forcing and all two-step iterations of the form ${\rm Add}(\kappa,\theta)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\lt\kappa$-closed and $\leq\kappa$-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 ${\rm GCH}$.