Hilbert on different notions of completeness: a conceptual and historical analysis

Abstract

It is typically recognized that Gödel’s negative results undermined Hilbert’s Program, at least in its strictest form. In this paper we point out the way in which Hilbert himself had created the possibility of Gödel doing so. We highlight the crucial role Hilbert played in the evolution and stabilization of the term “completeness”, and analyze the impact of his work on results obtained by researchers working close to him, but also by authors belonging to different traditions of mathematical practice, and whose views (that is, mathematical and philosophical views) on the nature of their field did not coincide with those of Hilbert; the major example here is Kurt Gödel. Furthermore, we show that an informal version of the modern notion of completeness is already presented by Hilbert as early as 1899. Our historiographical objective is thus twofold: on the one hand, to suggest a new way of showing what we mean when we say that the completeness theorem took about thirty years to be stated as such, and merely one year to be resolved; and on the other, to point out networks of exchange of ideas and the double nature of these ideas, as mathematical practices, and as (meta)mathematical or epistemological views.