Hilbert on categoricity and completeness

Abstract

The article analyzes Hilbert’s early contributions to the metatheoretic concept of categoricity and its relation to other completeness properties of axiomatic theories. For this purpose, we present a categoricity proof of the axiom system for real analysis first sketched in his lecture course Logische Prinzipien des mathematischen Denkens from 1905. This result will be compared with Dedekind’s well-known categoricity theorem for arithmetic from 1888 as well as with Hilbert’s informal remarks on the completeness of his axiom system for Euclidean geometry presented in Grundlagen der Geometrie (1899). Given these early metatheoretical results, we will address Hilbert’s motivations for the use of categoricity arguments in the foundations of mathematics.