Completeness, Categoricity and Imaginary Numbers: The Debate on Husserl

Abstract

<jats:p>Husserl's two notions of "definiteness" enabled him to clarify the problem of imaginary numbers. The exact meaning of these notions is a topic of much controversy. A "definite" axiom system has been interpreted as a syntactically complete theory, and also as a categorical one. I discuss whether and how far these readings manage to capture Husserl's goal of elucidating the problem of imaginary numbers, raising objections to both positions. Then, I suggest an interpretation of "absolute definiteness" as semantic completeness and argue that this notion does not suffice to explain Husserl's solution to the problem of imaginary numbers.</jats:p>

Completeness: From Husserl to Carnap

“…I have recently proposed that this sense of "absolute definiteness" should be understood as semantic completeness, rather than syntactic completeness, to make more plausible Husserl's solution to the problem of imaginary numbers (cf. [1]). Although (in 1901) the "standard cut" between provability and logical consequence had not yet been defined, I think that there are reasons to accept that Husserl's notion of consequence (explicitly given in the above quote) was basically semantical, at least in the context of absolute definiteness.…”

“…[6]), is one field of investigation. 1 A second important topic was the discussion on the writings of Huntington (cf. [33,34]) and Veblen (cf.…”

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