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

Abstract

Goal. We analyze when the Partition Principle ($\mathsf{PP}$) holds without $\mathsf{AC}$ in models arising from a free finite $H$-action on Cantor space, and reconcile two standard routes to such models. Approach. Route I proceeds via a Boolean-valued presentation $\mathrm{Sh}(\mathbb{B})$ and symmetric names; Route II uses direct forcing with $\mathrm{Fn}(\mathbb{N}\times H,2)$ and finite-support automorphisms. We prove a unification theorem identifying the resulting symmetric submodels and develop a Local-to-Global Embedding Principle (LEP) for hereditarily symmetric names. Results. We prove external $\mathsf{PP}$ in the symmetric model $N$ built via Route II. From LEP we obtain that $\mathsf{PP}$ holds in the symmetric model, hence $N\models \mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$. Along the way, we unify the Route I/Route II presentations functorially. Limitations. Our proof exploits the countable-support stratification of $\mathrm{Fn}(\mathbb{N}\times H,2)$; extending the LEP/gluing to uncountable presentations (e.g. $\mathrm{Fn}(\kappa\times H,2)$ for $\kappa>\omega$) or, more generally, to $\kappa$-directed families of finite supports remains open.