From Internal to External: Classical Models of ZF + PP + $\neg$AC

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 $\mathrm{Fn}(\mathbb{N}\times H,2)$. Those claims are withdrawn.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 $f\circ s={\mathrm{id}}_B$. 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.

arXiv commentMajor correction. The proposed internal-to-external construction is withdrawn. Its central LEP theorem constructs right inverses to all surjections, which is equivalent to AC and contradicts the claimed failure of AC. The localization, gluing, route-unification, and non-well-orderability arguments also fail; see the correction notice in the replacement manuscript