Take the 2-minute tour ×  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?  share|improve this question Haven't you
answered your own question? The compactness theorem implies that
if a theory $T$ has arbitrarily large finite models, then it has
an infinite model. So it gives a special case of upward
Löwenheim-Skolem, but as far as I can tell the argument doesn't
extend to larger cardinalities. –  Qiaochu Yuan Jun 15 '12 at
14:09 add comment  1 Answer  up vote 5 down vote accepted  Well,
one might say everyt