Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes

TL;DR

This paper constructs a topos from a group action on Cantor space to model set theory principles, demonstrating the Partition Principle holds while the Axiom of Choice fails internally.

Contribution

It introduces a novel method using symmetric iterations and sheaf-toposes to model set theories satisfying the Partition Principle without Choice.

Findings

The internal set universe satisfies the Partition Principle.

The model demonstrates failure of the Axiom of Choice internally.

Constructs a model of ZF + PP + ¬AC within a sheaf topos.

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