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From Internal to External: Classical Models of ZF + PP + $\neg$AC
Logic · Submitted 12 Nov 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It asserts no model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ and no consistency or independence result.
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The abstract, sentence by sentence
An earlier form of this manuscript claimed to obtain a classical symmetric model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ by two routes: a Boolean-valued externalization of an internal sheaf-topos construction, and a direct symmetric extension based on $\operatorname{Fn}(\mathbb{N}\times H,2)$. Those claims are withdrawn. The decisive contradiction is already present in the stated Local-to-Global Embedding Principle. For every hereditarily symmetric surjection $f\twoheadrightarrow B$, that theorem purports to construct a function $s\to A$ satisfying $f\circ s=\operatorname{id}_B$. Thus it asserts that every surjection has a right inverse, which is equivalent over $\mathsf{ZF}$ to the Axiom of Choice and is stronger than the injection required by the Partition Principle. It therefore cannot prove $\mathsf{PP}+\neg\mathsf{AC}$. The supporting localization and gluing arguments fail independently: a condition cannot be strengthened while deleting coordinates outside a fixed support; the fixed support is replaced in the proof by element-dependent supports; the proposed coherent antichains cannot exist below arbitrary Boolean values; minimality in an arbitrary ground-model well-order does not imply equivariance; right inverses need not agree on overlaps; and countably many finite supports need not combine into a finite support. The route-unification theorem and the proposed witness to $\neg\mathsf{AC}$ have further defects. This replacement retains only standard background facts about symmetric extensions and records the obstructions. It asserts no model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ and no consistency or independence result.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An earlier form of this manuscript claimed to obtain a classical symmetric model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ by two routes: a Boolean-valued externalization of an internal sheaf-topos construction, and a direct symmetric extension based on $\operatorname{Fn}(\mathbb{N}\times H,2)$. | background 0.77 | 0.68 | 0.04 | asserted 0.96 |
| s1 | Those claims are withdrawn. | limitation 0.87 | 0.81 | 0.26 | asserted 1.00 |
| s2 | The decisive contradiction is already present in the stated Local-to-Global Embedding Principle. | result 0.72 | 0.52 | 0.26 | asserted 1.00 |
| s3 | For every hereditarily symmetric surjection $f\twoheadrightarrow B$, that theorem purports to construct a function $s\to A$ satisfying $f\circ s=\operatorname{id}_B$. | result 0.38 | 0.23 | 0.07 | asserted 0.90 |
| s4 | Thus it asserts that every surjection has a right inverse, which is equivalent over $\mathsf{ZF}$ to the Axiom of Choice and is stronger than the injection required by the Partition Principle. | interpretation 0.61 | 0.26 | 0.15 | asserted 1.00 |
| s5 | It therefore cannot prove $\mathsf{PP}+\neg\mathsf{AC}$. | result 0.61 | 0.48 | 0.31 | asserted 1.00 |
| s6 | The supporting localization and gluing arguments fail independently: a condition cannot be strengthened while deleting coordinates outside a fixed support; the fixed support is replaced in the proof by element-dependent supports; the proposed coherent antichains cannot exist below arbitrary Boolean values; minimality in an arbitrary ground-model well-order does not imply equivariance; right inverses need not agree on overlaps; and countably many finite supports need not combine into a finite support. | result 0.50 | 0.59 | 0.07 | asserted 1.00 |
| s7 | The route-unification theorem and the proposed witness to $\neg\mathsf{AC}$ have further defects. | limitation 0.57 | 0.63 | 0.05 | asserted 0.99 |
| s8 | This replacement retains only standard background facts about symmetric extensions and records the obstructions. | contribution 0.47 | 0.70 | 0.18 | asserted 0.99 |
| s9 | It asserts no model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ and no consistency or independence result. | limitation 0.97 | 0.69 | 0.75 | asserted 0.98 |
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The paper
Original abstract
An earlier form of this manuscript claimed to obtain a classical symmetric model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ by two routes: a Boolean-valued externalization of an internal sheaf-topos construction, and a direct symmetric extension based on $\operatorname{Fn}(\mathbb{N}\times H,2)$. Those claims are withdrawn. The decisive contradiction is already present in the stated Local-to-Global Embedding Principle. For every hereditarily symmetric surjection $f\twoheadrightarrow B$, that theorem purports to construct a function $s\to A$ satisfying $f\circ s=\operatorname{id}_B$. Thus it asserts that every surjection has a right inverse, which is equivalent over $\mathsf{ZF}$ to the Axiom of Choice and is stronger than the injection required by the Partition Principle. It therefore cannot prove $\mathsf{PP}+\neg\mathsf{AC}$. The supporting localization and gluing arguments fail independently: a condition cannot be strengthened while deleting coordinates outside a fixed support; the fixed support is replaced in the proof by element-dependent supports; the proposed coherent antichains cannot exist below arbitrary Boolean values; minimality in an arbitrary ground-model well-order does not imply equivariance; right inverses need not agree on overlaps; and countably many finite supports need not combine into a finite support. The route-unification theorem and the proposed witness to $\neg\mathsf{AC}$ have further defects. This replacement retains only standard background facts about symmetric extensions and records the obstructions. It asserts no model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ and no consistency or independence result.
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