Completeness: From Husserl to Carnap

Abstract

In his Doppelvortrag (1901), Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the history of completeness. Some commentators defended that the meaning of these notions should be understood as categoricity, while other scholars believed that it is closer to syntactic completeness. A detailed study of the early twentieth-century axiomatics and Husserl’s Doppelvortrag shows, however, that many concepts of completeness were conflated as equivalent. Although “absolute definiteness” was principally an attempt to characterize non-extendible manifolds and axiom systems (different from Hilbert’s axiom of completeness), an absolutely definite theory has a unique model and, thus, it is non-forkable and semantically complete (decidable). Non-forkability and decidability were formally delimited by Fraenkel and Carnap almost three decades later and, in fact, they mentioned Husserl as precursor of the latter. Therefore, this paper contributes to a reassessment of Husserl’s place in the history of logic.