A countable-support symmetric iteration separating PP from AC
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908872873082880 |
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| author | Gilson, Frank |
| author_facet | Gilson, Frank |
| contents | We construct, from a ground model of $ZFC$, a transitive symmetric model $M$ satisfying $ZF + DC + PP + AC_{wo} + \neg AC$. The construction starts with a Cohen symmetric seed model $N$ over $Add(ω,ω_1)$ and performs an Ord-length countable-support symmetric iteration. For fixed parameters $S:=A^ω$ and $T:=PowerSet(S)$ (as computed in $N$), successor stages add orbit-symmetrized packages which force the localized splitting principle $PP^{\mathrm{split}}\!\restriction T$ (hence $PP\restriction T$) and the choice principle $AC_{wo}$, while preserving $DC$ and keeping $A$ non-well-orderable. A diagonal-lift/diagonal-cancellation scheme produces $ω_1$-complete normal limit filters. A persistence argument yields $SVC^+(T)$ in M, and Ryan--Smith localization then upgrades $PP\restriction T$ and $AC_{wo}$ to $PP$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01855 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A countable-support symmetric iteration separating PP from AC Gilson, Frank Logic 03E35 We construct, from a ground model of $ZFC$, a transitive symmetric model $M$ satisfying $ZF + DC + PP + AC_{wo} + \neg AC$. The construction starts with a Cohen symmetric seed model $N$ over $Add(ω,ω_1)$ and performs an Ord-length countable-support symmetric iteration. For fixed parameters $S:=A^ω$ and $T:=PowerSet(S)$ (as computed in $N$), successor stages add orbit-symmetrized packages which force the localized splitting principle $PP^{\mathrm{split}}\!\restriction T$ (hence $PP\restriction T$) and the choice principle $AC_{wo}$, while preserving $DC$ and keeping $A$ non-well-orderable. A diagonal-lift/diagonal-cancellation scheme produces $ω_1$-complete normal limit filters. A persistence argument yields $SVC^+(T)$ in M, and Ryan--Smith localization then upgrades $PP\restriction T$ and $AC_{wo}$ to $PP$. |
| title | A countable-support symmetric iteration separating PP from AC |
| topic | Logic 03E35 |
| url | https://arxiv.org/abs/2601.01855 |