 So matters remained unclear throughout the 1920s. But the constructivist ambitions of the Hilbert school, the focus on the analysis of the quantifiers, and the explicit posing of metalogical questions had made the emergence of first-order logic as a system worthy of study in its own right all but inevitable. The crucial technical breakthroughs came in 1929 and 1931 with the publication, by Gödel, first, of the completeness theorem for first-order logic, and then of the incompleteness theorems. With these results (and others that soon followed) it finally became clear that there were important metalogical differences between first-order logic and higher-order logics. Perhaps most significantly, first-order logic is complete, and can be fully formalized (in the sense that a sentence is derivable from the axioms just in case it holds in all models). First-order logic moreover satisfies both compactness and the downward Löwenheim-Skolem property; so it has a tractable model theory. Second-order logic does not. By the middle of the 1930s these distinctions were beginning to be widely understood, as was the fact that categoricity can in general only be obtained in higher-order systems. Lindström was later to show (1969) that no logical system satisfying both compactness and the Löwenheim-Skolem property can possess greater expressive power than first-order logic: so in that sense, first-order logic is indeed a “natural” entity.