Funny that no one mentioned it so far. I find the compactness theorem magical, actually:

A first order theory $T$ has a model if and only if every finite subset of $T$ has a model.

This let's you derive the finite form of Ramsey's theorem from the infinite. That is magic. for most applications of this kind, the compactness theorem for propositional calculus is actually enough.

The Compactness Theorem

Funny that no one mentioned it so far. I find the Compactness Theorem magical, actually:

A first order theory $T$ has a model if and only if every finite subset of $T$ has a model.

This let's you derive the finite form of Ramsey's theorem from the infinite form. That is magic. For most applications of this kind, the Compactness Theorem for propositional calculus is actually enough.

Compactness for first-order logic and propositional logic are actually equivalent (over ZF). In fact, there is a rather large collection of equivalent results:

• Boolean Prime Ideal Theorem
• The Ultrafilter Lemma
• The Stone Representation Theorem
• The Tychonoff Theorem for compact Hausdorff spaces

The Compactness Theorem is strictly weaker than the Axiom of Choice, but it is not provable in plain ZF. This is often used to show that certain results are weaker than the Axiom of Choice. For example, it can be used to show that the existence of non-measurable sets is weaker than the Axiom of Choice (by an old result of Sierpiński).