[pdonis]

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.

The Godel theorems apply to any first-order axiom system, regardless of whether it has an infinite model or not.

[Epa095]

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.

I 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.

[Paracompact]

He's right, the Godel theorems have nothing to do with existence of models satisfying this-or-that. Such would be "semantic" truths. The reason Hilbert's program survived Löwenheim–Skolem is that Hilbert was a formalist concerned with "syntactic" truth, that is, whether there are statements P such that neither P nor not-P could be proven by the axiom system.

You might think LS would trivially demonstrate as much---"Just take P = our underlying model is countable!"---but this is not formalizable within the system itself.