There is a passage in (Skolem 1928) that has generated multiple conflicting interpretations (see discussion in Van Heijenoort's intro to (Skolem, 1928) in [1967b, p. 508] as well as (Goldfarb, 1971). Skolem states what is now recognizable as a key lemma of the completeness theorem: either for some $n$ there is no solution of level $n$, or, for every $n$, there are solutions of level $n$. (“Solutions” are truth-value assignments to propositional instances approximating to a first-order formula in normal form - see https://mathoverflow.net/q/373381/116705 for background).
Instead of proving completeness by showing that the right-hand disjunct implies satisfiability (a direct consequence of the Lowenheim-Skolem theorem), Skolem gives a syntactic argument to show that “if there are solutions for arbitrarily high n, then the formula is (syntactically) consistent”. Conversely, if there is an $n$ for which there are no solutions, the formula “contains a contradiction”. The controversy stems from the fact that Skolem’s switch to syntax and apparent avoidance of the Lowenheim-Skolem theorem seems unmotivated and the argument he gives is inconclusive.
Of the interpretations that have been put forward, none take into account Skolem’s aim of proving decidability for all first-order formulas. This of course was later discovered to be impossible, and I’m inclined to think that the obscurity of the syntactic argument can be attributed to Skolem’s struggle with this destined-to-fail task.
The only interpretation I can think of that would support this is as follows:
Perhaps recognizing the difficulty of deciding the key lemma above for the case of formulas of any prefix, Skolem tries to show how it can be replaced by a syntactic counterpart with more hope of being decidable: either the formula is contradictory, or the formula is consistent. To do this he would need to show exactly what he purports to show: (a) if there is a level with no solutions then the formula is contradictory, (b) if for every $n$, there are solutions of level $n$ then the formula is consistent.
If he could then show that there was a decidable method for determining the consistency of a basic theory (essentially, propositional logic plus rule of substitution plus the functional form of the formula added as an axiom), this would also decide the alternative given by Godel's lemma, and from there the satisfiability of the formula via the LST.
Would logicians in 1928 have had any reason to consider syntactic consistency to be a more tractable candidate for a decidability proof?
References:
GOLDFARB, WARREN D [1971] Review of Skolem 1970. The Journal of Philosophy 68, pp. 520-530.
Skolem, Thoralf. [1928] On Mathematical Logic. English translation in van Heijenoort (ed.) [1967], pp. 508– 524.
VAN HEIJENOORT, JEAN (ED.) [1967b] From Frege to Gödel; a source book in mathematical logic, 1879-1931. Cambridge, Harvard University Press.