The goal of this note is to present a proof of the following fundamental result. A theory is said to be satisfiable iff there is a model of
Theorem 1 (Compactness) Let be a first order theory. Suppose that any finite subtheory is satisfiable. Then itself is satisfiable.
As indicated on the set of notes on the completeness theorem, compactness is an immediate consequence of completeness. Here I want to explain a purely semantic proof, that does not rely on the notion of proof.
The argument I present uses the notion of ultraproducts. Although their origin is in model theory, ultraproducts have become an essential tool in modern set theory, so it seems a good idea to present them here. We will require the axiom of choice, in the form of Zorn’s lemma.
The notion of ultraproduct is a bit difficult to absorb the first time one encounters it. I recommend working out through some examples in order to understand it well. Here I confine myself to the minimum necessary to make sense of the argument.