Completeness Theorems and the Separation of the First and Higher-Order Logic

With his 1929 thesis Gödel delivers himself to us almost fully formed. He gives in it a definitive, mathematical treatment of the completeness theorem; but he also declares himself philosophically, with a cluster of philosophical ruminations, some argumentation, and some slightly mysterious philosophical remarks, in this first example of the bold extraction of philosophical observations from mathematical facts, which would become Gödel’s modus operandi and professional trademark. These latter remarks were never included in the publication based on the thesis – an unfortunate excision, perhaps, as Gödel would not be inclined to publish such unbuttoned philosophical material until the 1940s with his contribution to the Schilpp volume on Russell, ∗Research partially supported by a grant NWO and grant 40734 of the Academy of Finland. 1The author thanks the NWO foundation for their support during the writing of this paper; Steve Awodey, Paolo Mancosu, Dana Scott and Wilfried Sieg for helpful comments and correspondence about the material in Gödel’s thesis. The author is especially grateful to Jouko Väänänen, for extensive discussions about first and higher order logic.