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

2024/12/07 by Ben-Neria, Omer, Kaplan, Eyal · 1 citation
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2412.05493

Abstract

We construct a model of the form L[A,U] that exhibits the simplest structural behavior of σ-complete ultrafilters in a model of set theory with a single measurable cardinal κ , yet satisfies 2^κ= κ⁺⁺. 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.

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