On BPI in Symmetric Extensions Part 1
Brian Ransom
TL;DR
This work develops a direct, Ramsey-theoretic approach to proving the Boolean Prime Ideal Theorem ($\mathrm{BPI}$) in choiceless models by introducing the filter extension property (FEP) and its symmetric-system generalization. It proves $\mathrm{BPI}$ in the generalized Cohen model $N(I,Q)$ for large index sets $I$ by adapting Harrington’s Halpern-Läuchli method, and extends the result to small $I$ through internal isomorphisms between forcing relations using the $\Sigma$ and $\Gamma$ maps. A dynamical framework based on the Ramsey property and its virtual extension, the virtual Ramsey property, is developed to show sufficiency of these conditions for $\mathrm{BPI}$ in both permutation models and symmetric extensions, with a nuanced discussion of when these properties hold or fail. The paper also connects these constructions to Blass’ results and to forcing-based transfers across models, offering a foundation for further exploration of $\mathrm{BPI}$ preservation under symmetric iterations and related questions. Overall, it provides a coherent, direct toolkit for establishing $\mathrm{BPI}$ in choiceless settings via stable, Ramsey-inspired extension properties and their dynamical counterparts.
Abstract
Historically, proofs of $\mathrm{BPI}$ in models without choice have relied on a contradiction framework that was introduced by Halpern. We introduce the filter extension property for permutation models and symmetric extensions, which formalizes the naïve approach to extend arbitrary filters to ultrafilters by repeatedly extending filters by minimal increments. We use this framework to give the first direct proof of $\mathrm{BPI}$ in the generalized Cohen model $N(I,Q)$ -- a model that adds a Dedekind-finite set of mutually $Q$-generic filters over a ground model $M\vDash\mathrm{ZFC}$. In the case that the index set $I$ is large, we adapt Harrington's proof of the Halpern-Läuchli theorem to prove the result. We then extend the results from Karagila and Schlicht to show that $I$ can be assumed to be large without loss of generality. The approach given by Harrington's proof is essentially dynamical, and we show that this technique can be used in permutation models to reprove a direction of Blass' theorem: that a dynamical condition called the Ramsey property is sufficient for $\mathrm{BPI}$ to hold in a permutation model. We then introduce a dynamical generalization of the Ramsey property called the virtual Ramsey property, which abstracts core features of our adaptation of Harrington's proof, and we prove that the virtual Ramsey property is sufficient for $\mathrm{BPI}$ to hold in a symmetric extension.