Godel has argued that Skolem's finitism was responsible for his failure to prove completeness, despite having all the components of a proof. One can challenge this argument with the objection that Skolem did in fact use non-finitary reasoning in his proof of the Lowenheim-Skolem theorem. The latter theorem requires applications of law of excluded middle to infinite sets. Godel addresses this challenge directly:
That he used non-finitary reasoning for Lowenheim’s Theorem proves nothing, because pure model theory, where the concept of proof does not come in, lies on the borderline between mathematics and metamathematics (Letter to Wang, in Wang, 1974, p. 10)
As Wang continues,
Skolem probably thought he could not use the same sort of argument when considering the question of completeness, which is squarely in the domain of metamathematics. (Wang, Letter to Godel, 19 December, 1967).
But what is the basis for Godel's claim that the completeness theorem, in contrast to the Lowenheim-Skolem theorem, belongs to the domain of metamathematics because it ``involves the notion of proof''?
If Godel had said "consistency" instead of "proof", then his response would make sense in the context of Hilbert's program. Hilbertian finitism holds that non-finitary reasoning is permitted provided it has been given a justification, namely, a consistency proof, in a finitary metatheory. Suppose that instead of completeness Godel was talking about a proof of the consistency of the theory in which one proves the Lowenheim-Skolem theorem. Then it would be true to say that a finitist would permit non-finitary reasoning only in proving the latter theorem - this reasoning would be justified in terms of the former consistency proof.
But the completeness theorem only involves consistency in a hypothetical sense: it shows that IF a formula (or theory) is consistent, then it is satisfiable. This does not seem to have bearing on the justification of the object theory in which we prove the Lowenheim-Skolem theorem, or of the law of excluded middle.
So, (1) if non-finitary uses of the law of excluded middle are unobjectionable for Skolem (and the finitist) in the context of the Lowenheim-Skolem theorem, why would they suddenly become objectionable in the context of completeness? (2) Why is completeness metatheoretical relative to the Lowenheim-Skolem theorem, if the theory whose completeness is in question is not the same as the theory in which the LST is proven?