First, two exercises to work some with the notion of ultrapower: Check that whenever is a nonprincipal ultrafilter on the natural numbers, and
- for all or
Our argument for compactness required the existence of nonprincipal ultrafilters. One might wonder whether this is a necessity or just an artifact of the proof. It is actually necessary. To see this, I will in fact show the following result as a corollary of compactness:
Theorem. If is a nonprincipal filter on a set then there is a nonprincipal ultrafilter on that extends
(Of course, this is a consequence of Zorn’s lemma. The point is that all we need is the compactness theorem.)
Proof. Consider the language Here, each is a constant symbol, is another constant symbol, and is a symbol for a binary relation (which we will interpret below as membership).
In this language, consider the theory A model of this theory would look a lot like except that the natural interpretation of in namely, is no longer nonprincipal in , because is a common element of all these sets.
Note that there are indeed models of thanks to the compactness theorem.
If let and note that is a nonprincipal ultrafilter over that contains