Iterating PP-packages without Choice: A Cohen symmetric seed and a localization framework
Frank Gilson
TL;DR
The paper proves that the Partition Principle $\mathsf{PP}$ does not imply the Axiom of Choice by constructing a transitive model of $\mathsf{ZF}+\mathsf{DC}+\mathsf{PP}+\neg\mathsf{AC}$. It starts with a Cohen symmetric seed model $\mathcal{N}$ built over $\mathop{Add}(\omega,\omega_1)$ and then performs a class-length countable-support symmetric iteration that localizes $\mathsf{PP}$ to $T=\mathcal{P}(S)$ while forcing $\mathsf{AC}_{\mathsf{WO}}$, preserving $\mathsf{DC}$ and maintaining a non-well-orderable witness $A$. Ryan–Smith localization is used to show $\mathsf{PP}$ is equivalent to $\mathsf{PP}\restriction T \wedge \mathsf{AC}_{\mathsf{WO}}$ under $\mathsf{SVC}^{+}(T)$, so the local $\mathsf{PP}$ and $\mathsf{AC}_{\mathsf{WO}}$ suffice for global $\mathsf{PP}$. The construction introduces orbit-symmetrized packages and diagonal-cancellation automorphisms to achieve stable, hereditarily symmetric generics, and a modified limit filter to control conjugation across stages. The resulting model satisfies $\mathsf{ZF}+\mathsf{DC}+\mathsf{PP}$ but not $\mathsf{AC}$, advancing the understanding of the independence between these set-theoretic principles and clarifying how local-to-global PP can be achieved without the full Axiom of Choice.
Abstract
The Partition Principle $\mathsf{PP}$ asserts that whenever there is a surjection $A\twoheadrightarrow B$, there is an injection $B\hookrightarrow A$. Russell conjectured in 1906 that $\mathsf{PP}$ is equivalent to the Axiom of Choice $\mathsf{AC}$; while $\mathsf{AC}\Rightarrow \mathsf{PP}$ is immediate, the converse has remained open. We show that $\mathsf{PP}$ does not imply $\mathsf{AC}$ by constructing a transitive model of $\mathsf{ZF}+\mathsf{DC}+\mathsf{PP}+\neg\mathsf{AC}$. Starting from a Cohen symmetric model $\mathcal{N}$ of $\mathrm{Add}(ω,ω_1)$ built with a countable-support symmetry filter, we fix parameters $S:=A^ω$ and $T:=\mathcal{P}(S)$ and perform a class-length countable-support symmetric iteration. At successor stages we use orbit-symmetrized packages that split targeted surjections, yielding $\mathsf{PP}\!\restriction T$ and $\mathsf{AC}_{\mathsf{WO}}$, while preserving $\mathsf{DC}$ and ensuring that $A$ remains non-well-orderable. A diagonal-cancellation/diagonal-lift infrastructure supplies a proper $ω_1$-complete normal filter at limit stages. Finally, Ryan--Smith localization shows that under $\mathsf{SVC}^+(T)$, $\mathsf{PP}$ is equivalent to $\mathsf{PP}\!\restriction T \wedge \mathsf{AC}_{\mathsf{WO}}$, so the final model satisfies $\mathsf{PP}$ but not $\mathsf{AC}$.