Before 1930, a number of logicians exhibited interest in, and tried to solve, the decision problem for first order logic without showing any similar interest in the completeness theorem. Specifically, Lowenheim (1915) and Skolem (1922, 1929) considered the implications of the Lowenheim-Skolem theorem for giving a decision procedure for fragments of FOL but show no recognition of the adjacent question of the completeness of their proof procedures.
This seems surprising. A decision procedure for FOL (later discovered to be impossible) would give a finite method of demonstrating whether a first-order formula was or was not satisfiable. Obviously, if we had such a procedure, it would entail that every formula was either satisfiable or demonstrably unsatisfiable, which is equivalent to completeness (at least in an informal sense, depending on the nature of the "demonstration").
But even this informal sense of completeness is ignored by Lowenheim and Skolem, despite being easily derivable from their proofs of the Lowenheim-Skolem theorem.
Explanations often point to the algebraists' greater interest in (what we now consider) model-theoretic vs. proof-theoretic questions. But what explains this greater interest?
In the algebraic framework, the question of the decidability of formulas was investigated as the question: does an equation have a solution?
My question is: are there specifically algebraic and/or technical considerations that could explain why decidability in this sense, but not completeness (even in an informal sense), was of interest to algebraists like Skolem and Lowenheim?
References:
LÖWENHEIM, LEOPOLD
[1915] On Possibilities in the Calculus of Relatives, in van Heijenoort (ed.) [1967], pp. 228–251.
SKOLEM, THORALF
[1922] Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre. Matematikerkongressen i Helsingfords den 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse . Helsinki: Akademiska Bokhandeln, 1923, pp. 217–232. English translation in van Heijenoort (ed.) [1967], pp. 290–231.
[1928] Über die mathematische Logik. Norsk matematisk tidsskrift , v. 10, pp. 125–142. English translation in van Heijenoort (ed.) [1967], pp. 508– 524.
[1929] Über einige Grundlagenfragen der Mathematik. Skrifter utgitt av Det Norske Videnskaps–Akademi i Oslo, I. Matematisk-naturvidenskapelig klasse , no. 4, pp. 1–49.
VAN HEIJENOORT, JEAN (ED.)
[1967] From Frege to Gödel; a source book in mathematical logic, 1879-1931. Cambridge, Harvard University Press.