The Ultrafilter Lemma and Its Place in Mathematics - Righard,Jon (Bachelor's Theses in Mathematical Sciences, 2024)
We investigate the strength of the Ultrafilter Lemma in comparison to the Axiom of Choice. In particular, we prove that a number of classical mathematical theorems are equivalent to the Ultrafilter Lemma over $\sf ZF$, and we study a few results which are typically only obtained in $\sf ZFC$ (as opposed to $\sf ZF$) but which can also be proven from the conjunction of $\sf ZF$ with the Ultrafilter Lemma. We finish with a foray into model theory, motivated by a desire to understand the notion of an ultraproduct.
Keywords: set theory; ultrafilter lemma; mathematical logic; foundations; topology; model theory