On BPI in Symmetric Extensions Part 1

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 na\"ive 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-L\"auchli 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.