UC Irvine

This thesis aims to address three main objectives: to better understand when, and how BPI holds in permutation models and in symmetric extensions; to develop a unifying relationship between variants of the Halpern-Läuchli theorem and BPI in symmetric extensions; to develop and expose simple dynamical perspectives on BPI in models without choice. Our approach to these questions is to introduce a characterization program, where we aim to provide conditions that are equivalent to BPI in permutation models and in symmetric extensions. By reproving known results, reworking known proofs, and proving new ones, we show how each of these characterizing properties offer novel perspectives on BPI, and how they address the three objectives listed above.The case of BPI in permutation models is studied in chapter 3. This work reproves and extends several results from [Bla87], [KPT05], and [Bla11]. These characterization theorems, and the proofs we offer to relate them, can be seen as simpler, motivating versions of the main theorems of this thesis, in chapter 4.In chapter 4, we characterize BPI in symmetric extensions. The characterizing properties that we introduce are based on the various properties studied in chapter 3 for permutation models, and on our new proof of BPI in the generalized Cohen model. This new proof is given in section 4.2, and we adapt methods from Harrington's forcing proof of the Halpern-Läuchli theorem.In chapter 5, we extend results from [KS20], and thereby answer a question asked in that paper. In the context of our characterization program, these results allow us to infer the general case of BPI in the generalized Cohen model from the particular case that Harrington's forcing proof can be adapted to handle.A concise description of the main finding in this work is that to extend symmetric filters to symmetric ultrafilters, one can consult an arbitrary (and not necessarily symmetric) ultrafilter in an outer model of choice. One can dynamically identify, inside this particular ultrafilter, all the information that is needed to extend the symmetric filter to a symmetric ultrafilter. We show that BPI holds in permutation models if and only if it can be proven in this manner.