A filter (in the sense of cjeris' writeup over there) E on a set X is called an ultrafilter if it is "maximal":

X admits no filter F such that E is a strict subset of F.

A principal ultrafilter Ex (for some x∈X) is a trivial example of an ultrafilter. If X is finite then it's obvious that no other ultrafilters exist. What if X is infinite?

Well, using Zorn's lemma (or the equivalent Hausdorff maximality lemma or Axiom of Choice, take your foundational pick) we can show that other (non-principal) ultrafilters exist.

A correspondence exists between the objects created in non-standard analysis and ultrafilters.

What ARE ultrafilters? A filter on X is one way of describing a classification of "almost all" elements of X. Say of A⊆X that it consists of "almost all" elements of X if A∈E. Note that the concepts "almost everywhere" (or "almost always") on a set of nonzero measure and of "second category" on a complete metric space give rise to filters: The filter of all sets with complement in X of measure zero, and the filter of all sets with complement of first category. So any filter can be treated as a generalisation of these concepts. But both of the original concepts have problems: not every set is measurable, and the topological concepts of Baire category theory are not necessarily applicable to any set. Any filter which is not an ultrafilter leaves similar "holes". By transforming our interest to an ultrafilter, we ensure that no holes are left.

Filters also occur in other places in mathematical logic, for instance in the construction of ultraproducts of models and their use. Los' theorem is the basis for work here. And interesting results about how first order logic can describe its world can be obtained. These often have no need to mention "ultrafilters"...

Yet more uses occur in algebra and analysis, of course.

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