Work

Hilbert on categoricity and completeness

Year: 2025

Type: article

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.

Source: Synthese

DOI: 10.1007/s11229-025-05345-4

Authors: Eduardo N. Giovannini, Georg Schiemer

Institutions: Consejo Nacional de Investigaciones Científicas y Técnicas, Universidad Nacional del Litoral, University of Vienna

Language: English

Topic: History and Theory of Mathematics

Subfield: Theoretical Computer Science

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funders: European Commission, Austrian Science Fund, Universität Wien, HORIZON EUROPE European Research Council