Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL?
I was thinking of another post of mine "Why accept the axiom of infinity?" when I though, "Well, if someone accepts arbitrary large finite numbers, what stops them from the jumping into the infinite?"
After all, doesn't compactness imply that if you have a theory $T$ with arbitrarily large finite models, then $T$ has arbitrarily large models—infinite or finite? This sounds a lot like the upward Lowenheim-Skolem theorem to me.
Is there a connection?
Well, one might say everything is connected, so it depends on which kind of connections you're looking for.
There certainly is the connection that one can derive upward Löwenheim-Skolem from downward Löwenheim-Skolem and compactness:
Let $T$ be a theory with equality that admits some infinite normal model $M$. For any $\aleph_\beta \ge |T|$ we can construct a model of cardinality $\aleph_\beta$ in two steps. (1) Construct a new theory by adjoining $\aleph_\beta$ new constant letters with axioms that say they are all different. By compactness, the extended theory is consistent and so has a model $N$. Clearly this model must have cardinality at least $\aleph_\beta$, and in particular it is a model of $T$. (2) If $|N|>\aleph_\beta$, use downward Löwenheim-Skolem to cut it down to size as a model of $T$.