Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915749061197824 |
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| author | Gilson, Frank |
| author_facet | Gilson, Frank |
| contents | We study the topos $\mathcal{E}=\mathsf{Sh}(H\ltimes 2^{\mathbb{N}})$ arising from a nontrivial finite group $H$ acting freely on Cantor space. Using a local embedding property for the relevant epimorphisms together with effective descent for monomorphisms, we show that the \emph{internal} set universe $V$ obtained from algebraic set theory (AST) inside $\mathcal{E}$ satisfies the Partition Principle. On the other hand, the quotient $q:X\to X/H$ is a small epimorphism in $\mathcal{E}$ with no section, and this yields (via the display interpretation) an internal surjection in $V$ with no internal section; hence $V\models\neg\mathsf{AC}$. In summary, $\mathcal{E}$ contains an internal model of $\mathsf{IZF}+\mathsf{PP}+\neg\mathsf{AC}$ (and if $\mathcal{E}$ is Boolean, equivalently after $\neg\neg$-sheafification, this upgrades to $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07675 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes Gilson, Frank Logic Category Theory 03E25, 03E35, 18B25 We study the topos $\mathcal{E}=\mathsf{Sh}(H\ltimes 2^{\mathbb{N}})$ arising from a nontrivial finite group $H$ acting freely on Cantor space. Using a local embedding property for the relevant epimorphisms together with effective descent for monomorphisms, we show that the \emph{internal} set universe $V$ obtained from algebraic set theory (AST) inside $\mathcal{E}$ satisfies the Partition Principle. On the other hand, the quotient $q:X\to X/H$ is a small epimorphism in $\mathcal{E}$ with no section, and this yields (via the display interpretation) an internal surjection in $V$ with no internal section; hence $V\models\neg\mathsf{AC}$. In summary, $\mathcal{E}$ contains an internal model of $\mathsf{IZF}+\mathsf{PP}+\neg\mathsf{AC}$ (and if $\mathcal{E}$ is Boolean, equivalently after $\neg\neg$-sheafification, this upgrades to $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$). |
| title | Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes |
| topic | Logic Category Theory 03E25, 03E35, 18B25 |
| url | https://arxiv.org/abs/2511.07675 |