Zermelo s1931b

Abstract

Through Skolem 1930 Zermelo became aware of Thoralf Skolem’s result (1923) that first-order set theory, if consistent, admits a countable model. Viewing axiomatic set theory as a foundation of Cantorian set theory with its unlimited progression of infinite cardinalities, Zermelo flatly rejected the basis of Skolem’s argument, the first-order formulation of the axioms of separation and replacement. In order to overcome the weakness of first-order logic, he started to realize a program that is charted in his theses concerning the infinite in mathematics, s1921: the development of infinitary languages and an infinitary logic as a means of giving mathematics a foundation which would preserve its “true” character. A second line he pursued consisted in developing set theory without Skolem’s limitations on separation, namely by “keeping with the true spirit of set theory, [allowing] for the free division, and [postulating] the existence of all [subsets] formed in an arbitrary way” (s1930d); this resulted in his second-order axiom system of set theory and the cumulative hierarchy as developed in his 1930a.