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

TL;DR

This paper investigates models of set theory where the Partition Principle holds without the Axiom of Choice, using two different forcing approaches and establishing a unification and embedding principle.

Contribution

It unifies two standard forcing routes to models with PP without AC and develops a Local-to-Global Embedding Principle for symmetric names.

Findings

Proves external PP in symmetric models built via finite-support automorphisms.

Establishes that PP holds in the symmetric model N, which satisfies ZF + PP + ¬AC.

Unifies the two forcing presentations functorially.

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.