The Hartogs-Lindenbaum Spectrum of Symmetric Extensions
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909289589768192 |
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| author | Ryan-Smith, Calliope |
| author_facet | Ryan-Smith, Calliope |
| contents | We expand the classic result that $\mathsf{AC}_{\mathsf{WO}}$ is equivalent to the statement "For all $X$, $\aleph(X)=\aleph^*(X)$" by proving the equivalence of many more related statements. Then, we introduce the Hartogs-Lindenbaum spectrum of a model of $\mathsf{ZF}$, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of $\mathsf{ZFC}$. We prove that all such spectra fall into a very rigid pattern. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Hartogs-Lindenbaum Spectrum of Symmetric Extensions Ryan-Smith, Calliope Logic 03E25 (Primary) 03E65 (Secondary) We expand the classic result that $\mathsf{AC}_{\mathsf{WO}}$ is equivalent to the statement "For all $X$, $\aleph(X)=\aleph^*(X)$" by proving the equivalence of many more related statements. Then, we introduce the Hartogs-Lindenbaum spectrum of a model of $\mathsf{ZF}$, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of $\mathsf{ZFC}$. We prove that all such spectra fall into a very rigid pattern. |
| title | The Hartogs-Lindenbaum Spectrum of Symmetric Extensions |
| topic | Logic 03E25 (Primary) 03E65 (Secondary) |
| url | https://arxiv.org/abs/2309.12100 |