A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis

Omer Ben-Neria; Eyal Kaplan

A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis

Logic 2024-12-10 v1

Authors: Omer Ben-Neria , Eyal Kaplan

Abstract

We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$ -complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$ . This result establishes a limitation on the extent to which structural properties of ultrafilters can determine the cardinal arithmetic at large cardinals, and answers a question posed by Goldberg concerning the failure of the Continuum Hypothesis at a measurable cardinal in a model of the Ultrapower Axiom. The construction introduces several methods in extensions of embeddings theory and fine-structure-based forcing, designed to control the behavior of non-normal ultrafilters in generic extensions.

Keywords

Kunen model measurable cardinal ultrafilter continuum hypothesis fine structure

Cite

@article{arxiv.2412.05493,
  title  = {A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis},
  author = {Omer Ben-Neria and Eyal Kaplan},
  journal= {arXiv preprint arXiv:2412.05493},
  year   = {2024}
}

A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis

Abstract

Keywords

Cite

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