A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem).


Two years later I don't know why I wrote the above note but I think it was because I wanted to prove the consistency of the  idea of Godel that a being can have an infinity of positive traits. He uses this idea in his proof of the existence of God. And it has been criticized. I thought to answer that criticism with the Lowenheim Skolem theorem. [Say the ''s'' in Skolem as in English-- not ''sh''.][This is similar to why C^n is not used instead of C^infinity, which is the it makes no difference which manifold you use. So C^infinity is always used.]

The actual idea of Gödel's proof of the existence of God came from Anselm. People argued whether it was rigorous or not. Leibniz showed that it is. Godel put it into logical notation and thus it is easily shown by on proof checking software program computer that is is a rigorous proof. No one dares to suggest that the proof is not rigorous. Rather the critics focus on the axioms.

[Hegel held that it is a valid proof and in fact the entire system oif Hegel is all about how to get to God--the absolute- as he makes clear.]