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 (Schindler in Bull Symbolic Logic 6(2):176–184, 2000). 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 <κ-closed ≤κ-distributive forcing and all two-step iterations of the form \({Add(\kappa,\theta)*\dot{\mathbb R}}\), where \({\dot{\mathbb R}}\) is forced to be <κ-closed and ≤κ-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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Correspondence to Victoria Gitman.

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Cheng, Y., Gitman, V. Indestructibility properties of remarkable cardinals. Arch. Math. Logic 54, 961–984 (2015). https://doi.org/10.1007/s00153-015-0453-8

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Keywords

  • Large cardinals
  • Remarkable cardinals
  • Indestructibility
  • Laver functions

Mathematics Subject Classification

  • 03E35
  • 03E55