Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes

Abstract

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 $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$).