Ultrafilters as Propositional Theories

This paper presents a philosophically illuminating explanation of the concepts, from mathematics and formal model theory, of filters and ultrafilters. If a propositional theory is just a set of propositions, then a filter is a propositional theory that is (i) consistent, (ii) closed under finite conjunction, and (iii) closed under implication. An ultrafilter is a filter that is also negation-complete. I prove the central theorem on ultrafilters and explain how it is a propositional variant of Lindenbaum’s Lemma (which says that every consistent set of sentences can be expanded to a negation-complete, consistent set). Next I introduce the idea of products and ultraproducts of filters and ultrafilters, which are models constructed from the filters or ultrafilters, and prove Łos’s theorem, which shows that the ultraproduct of an ultrafilter makes true all and only the sentences which express propositions in the ultrafilter.