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 $κ$ 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 $\leqκ$-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 ${\rm GCH}$.