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?