The Lowenheim Skolem theorem only applies to first-order axiom systems that have an infinite model. So it would apply to the axioms for the natural numbers, yes.
replyThe Godel theorems apply to any first-order axiom system, regardless of whether it has an infinite model or not.
I don't understand what you mean by this. Gödels two incompleteness theorems are about theories of natural numbers, so their models are infinite. I don't understand what you could mean by them applying to finite models.
replyI stand by my claim. The key point of Gödels incompleteness is NOT that no single theory can pin down a single model, that was known before.