Hilbert, completeness and geometry (Q2926205)

This is a rather lengthy discussion of Hilbert's axiom of completeness in his \textit{Grundlagen der Geometrie}. Much is made of Hilbert's emphasis on the introduction of numbers by geometric means. The author reaches the conclusion that ``Hilbert's analysis of the notion of continuity led him to formalize the Axiom of Completeness as a sufficient condition for analytic geometry, in the form of a maximality principle.''NEWLINENEWLINEHe then searches (without citing the most important paper on this subject [\textit{R. Carnap} and \textit{F. Bachmann}, Erkenntnis 6, 166--188 (1936; Zbl 0015.04902)]) for other examples of such axioms, and finds that the Church-Turing thesis is its analog in the theory of computability, and that the axiom of induction as a second-order principle is another such example in the axiomatization of arithmetic by Peano and Dedekind.