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 $${L(\mathbb R)}$$L(R) is absolute for proper forcing (Schindler in Bull Symbolic Logic 6(2):176---184, 2000). Here, we study the indestructibility properties of remarkable cardinals. We show that if ? is remarkable, then there is a forcing extension in which the remarkability of ? becomes indestructible by all <?-closed ≤?-distributive forcing and all two-step iterations of the form $${Add(\kappa,\theta)*\dot{\mathbb R}}$$Add(?,?)?R?, where $${\dot{\mathbb R}}$$R? is forced to be <?-closed and ≤?-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 GCH.
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