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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.07866 |
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| _version_ | 1866915749077975040 |
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| author | Gilson, Frank |
| author_facet | Gilson, Frank |
| contents | We develop a unified framework for iterated symmetric extensions with countable support and, more generally, with $<κ$-support. Set-length iterations are treated uniformly, and when the iteration template is first-order definable over a Godel-Bernays ground with Global Choice, the construction extends to class-length iterations. At limit stages with $\mathrm{cf}(λ)\geκ$ we use direct limits; when $\mathrm{cf}(λ)<κ$ we use inverse-limit presentations via trees of conditions together with tuple-stabilizer symmetry filters. The resulting limit filters are normal and $κ$-complete, yielding closure of hereditarily symmetric names and preservation of $\mathrm{ZF}$. Under a $κ$-Baire (strategic closure) hypothesis we obtain $DC_{<κ}$, and under a Localization hypothesis we obtain $DC_κ$. In the countable-support setting we give an $ω_1$-length construction adding reals and refuting $\mathrm{AC}$ while preserving $\mathrm{ZF}+\mathrm{DC}$, and we treat mixed products via stable pushforwards and restrictions. For singular $κ$, we develop the $\mathrm{cf}(κ)=ω$ case using block-partition stabilizers and trees; for arbitrary singular $κ$ we introduce game-guided fusion of length $\mathrm{cf}(κ)$ and a tree-fusion master condition, obtaining singular-limit completeness, preservation of $DC_{<κ}$, no collapse of $κ$, and no new subsets of any $λ<κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetric Iterations with Countable and $<κ$-Support: A Framework for Choiceless ZF Extensions Gilson, Frank Logic 03E35 We develop a unified framework for iterated symmetric extensions with countable support and, more generally, with $<κ$-support. Set-length iterations are treated uniformly, and when the iteration template is first-order definable over a Godel-Bernays ground with Global Choice, the construction extends to class-length iterations. At limit stages with $\mathrm{cf}(λ)\geκ$ we use direct limits; when $\mathrm{cf}(λ)<κ$ we use inverse-limit presentations via trees of conditions together with tuple-stabilizer symmetry filters. The resulting limit filters are normal and $κ$-complete, yielding closure of hereditarily symmetric names and preservation of $\mathrm{ZF}$. Under a $κ$-Baire (strategic closure) hypothesis we obtain $DC_{<κ}$, and under a Localization hypothesis we obtain $DC_κ$. In the countable-support setting we give an $ω_1$-length construction adding reals and refuting $\mathrm{AC}$ while preserving $\mathrm{ZF}+\mathrm{DC}$, and we treat mixed products via stable pushforwards and restrictions. For singular $κ$, we develop the $\mathrm{cf}(κ)=ω$ case using block-partition stabilizers and trees; for arbitrary singular $κ$ we introduce game-guided fusion of length $\mathrm{cf}(κ)$ and a tree-fusion master condition, obtaining singular-limit completeness, preservation of $DC_{<κ}$, no collapse of $κ$, and no new subsets of any $λ<κ$. |
| title | Symmetric Iterations with Countable and $<κ$-Support: A Framework for Choiceless ZF Extensions |
| topic | Logic 03E35 |
| url | https://arxiv.org/abs/2511.07866 |