A filter (in cjeris' sense) is a useful tool for topology and various other branches of mathematics. In ultrafilter, ariels claims that using Zorn's lemma one can show that "non-principal" ultrafilters exist. Elsewhere, ariels uses the cofinite filter and extends it to a "non-principal" ultrafilter. This is a proof that this technique is valid. Of course, it applies to the boring "principal" ultrafilters as well.

For clarity's sake: lowercase letters are elements of sets, uppercase letters are sets, bold lowercase letters are filters (sets of sets), and bold uppercase letters are sets of filters (sets of sets of sets).

To be a filter, a set of sets must satisfy three properties (adapted from filter):

1. A filter cannot be empty.
2. Filters are closed under containment; that is, if A is in a filter, and B is a superset of A, then B is also in the filter.
3. Filters are closed under intersection; that is, if A and B are both in the filter, so is A ∩ B.

Ultrafilters satisfy one final property: in addition to the above, they are "maximal". It is easy to show that a maximal filter over a set X contains, for every subset A, either A or its complement in X. (If it didn't, after all, there would be a bigger filter that did and contradictions would crop up.)

If X is a finite set, the only ultrafilters about are the "principal" ones that are generated by an element of the set. For example, if X = {1, 2, 3}, then the principal ultrafilter for 2 is a₂ = { {2}, {1, 2}, {2, 3}, {1, 2, 3} }. In this case, it's easy to see that filters can be extended to ultrafilters. The filter {{2, 3}, {1, 2, 3}} can be extended to a₂ above, or also to the principal ultrafilter generated by 3.

But finite sets are boring. Let X be an infinite set. Now we're playing with fire! If we want non-principal ultrafilters, we have to eschew finite sets, or else the filter will extend to a principal ultrafilter. Luckily, there is a cofinite filter over X, the filter { A : X\A is finite }. Some set algebra shows that this is indeed a filter. If we extend it to an ultrafilter, that ultrafilter isn't going to be a principal one. This leads to formulating the following lemma:

Lemma: Let X be a set. If f is a filter over X, it can be extended to an ultrafilter.

An rough outline of this proof is given in [1], p.202. I've filled in the gaps, and made the motivation more explicit.

This proof follows the standard sort of zornification techniques that prove the existence of maximal objects. Zorn's lemma says: if every chain in a poset has an upper bound, then there exists a maximal element. It's equivalent to the infamous axiom of choice, and is wholly non-constructive. We're going to construct an ultrafilter with it.

Order the filters of X by containment, so that a is "bigger" than b iff a ⊆ b. This makes the collection of filters into a poset. With this order we can construct the set F of all filters bigger than f. Let C be any subset (i.e., chain) of F, inheriting F's order. To apply the lemma, we must find a filter that is an upper bound for C with respect to the ordering of F. I claim the union of the elements of C, u = ∪_c∈C c, is such a filter.

First, that u is a filter:

1. f is the smallest element of F, so clearly f is in the union of things bigger than it. u is not empty.
2. Let B be any set in u. If it is in u, B is present in at least one filter in C. Because all of B's supersets are present in that filter, they are present in u.
3. Let A and B be sets in u. Following the logic above, A is in some filter a in C and B is in b. WLOG, assume a is bigger than b, so that A is in b too. Within b, A ∩ B is present — b is a filter, after all. So A ∩ B is in u.

Also, in accordance with Zorn's lemma, u is an upper bound for C. This is obvious, since it contains than any filter in C. By Zorn's lemma, then, f is contained in a maximal filter (that is, an ultrafilter) over X. ◊

Applying this lemma about ultrafilters to the cofinite filter mentioned earlier, we get a strange beast. It has no finite sets, so it cannot be a principal ultrafilter. It is a non-principal ultrafilter, a very scary, non-intuitive thing indeed.

References:
[1] A First Course in Topology, Robert A Conover.