502 – Compactness

First, two exercises to work some with the notion of ultrapower: Check that whenever is a nonprincipal ultrafilter on the natural numbers, and

1. for all or
2. 

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

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