Introductory note to s1930d

For the three years 1929 through 1931 Zermelo was supported by a fellowship provided by the Notgemeinschaft der Deutschen Wissenschaft, the Emergency Association of German Science. This association was founded in 1920 by the leading German academies and scientific organizations in order to avert a collapse of research in the economically disastrous time after the First World War. With a Swiss pension meager and inadequate, Zermelo had in 1929 applied for the fellowship with a research project “On the Nature and Foundations of Pure and Applied Mathematics and the Significance of the Infinite in Mathematics”, and surely the fellowship was a significant factor in his return to the fray of foundational work. s1930d was a report, dated 3 December 1930, to the Notgemeinschaft for the continuation of the fellowship into 1931.1 Written for the purpose of apprising and explaining, the report is an informal account, well worth reading, of Zermelo’s early career as well as of his 1930a work and what was to be subsequent investigations (cf. s1930e). In what follows, we selectively make a few of the several possible points that can be made. Zermelo makes informative remarks about the two axioms newly adopted in 1930a. About the replacement axiom Zermelo writes: “. . . [the 1908b axiom system] has received a valuable addition only once in the form of the ‘replacement axiom’ proposed by Fraenkel. On the other hand, it was only the concept of ‘cofinality’ introduced by Hausdorff that made possible a fruitful application of the new axiom.” How Hausdorff had first ventured beyond Cantor into the higher uncountable is suitably noted; today however, one would mainly regard replacement as the provenance of formalized definitions by transfinite recursion and the sets they provide. As for the axiom of foundation, after articulating the problems “posed and solved” in his 1930a, Zermelo newly highlights the axiom: “Using the new axiom it was possible to carry out a decomposition into layers, the ‘development’ of a ‘normal domain’ (that is, of a domain of sets satisfying the axiom system), and to answer the decisive main question in the ‘isomorphism theorems’.” Having described how, in his 1930a context, normal domains with equivalent bases are isomorphic to initial segments of each other, Zermelo writes: “From this already follows . . . that Cantor’s (generalized) conjecture [the generalized continuum hypothesis] . . . does not depend on the choice of the model, but that it is decided (as true or false) once and for all by means of our axiom system.” As Zermelo was evidently proceeding in second-order terms, this distinctly anticipates Georg Kreisel’s observation (cf. Kreisel 1971 ) that