A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915052824559616 |
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| author | Ben-Neria, Omer Kaplan, Eyal |
| author_facet | Ben-Neria, Omer Kaplan, Eyal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05493 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis Ben-Neria, Omer Kaplan, Eyal Logic 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. |
| title | A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis |
| topic | Logic |
| url | https://arxiv.org/abs/2412.05493 |