Zoom Talk: Eduardo Giovannini (Vienna)
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Hilbert’s Early Metatheory Revisited: Categoricity and Interpretability
This talk aims to provide a historically sensitive discussion of Hilbert’s
axioms of completeness and the early metatheory of formal axiomatics that
served as its logical foundation. This mathematical axiom is often regarded as one of his most original contributions to the development of the “model-theoretic” viewpoint in modern logic. First, we examine Hilbert’s understanding of mathematical languages and their interpretations; in particular, we argue that his early semantic views were informed by a particular notion of isomorphism. Second, we analyze Hilbert’s reflections on the central concept of categoricity. For this purpose, we study a categoricity proof of the axiom system for real analysis sketched by Hilbert in the lecture course Logische Prinzipien des mathematischen Denkens from 1905. Finally, we offer formal reconstructions of the axiom of completeness and investigate its logical connections with several notions of completeness of an axiom system.