Indestructibility properties of remarkable cardinals

TL;DR

This paper explores the indestructibility of remarkable cardinals under various forcing extensions, introduces the concept of a remarkable Laver function, and demonstrates the preservation of remarkability through GCH forcing.

Contribution

It establishes the indestructibility of remarkable cardinals under specific forcing notions and introduces the remarkable Laver function, advancing understanding of their robustness.

Findings

Remarkability can be made indestructible by certain forcing extensions.

Every remarkable cardinal admits a remarkable Laver function.

Remarkability is preserved by the canonical GCH forcing.

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 ${\rm GCH}$.