Zermelo s1931f

pp. 516-523

in: Ernst Zermelo, Set theory, miscellanea / Mengenlehre, varia, Berlin, Springer, 2010

Abstract

In his Warsaw talks s1929b Zermelo had proposed a new concept of set with the intention of separating sets from classes in a way more coherent than that of von Neumann. Without providing details, Zermelo takes as sets those classes which are domains of structures allowing for a categorical definition, i. e., a definition which up to isomorphism has exactly one model. There is no explanation about the additional relations allowed for definitions; moreover, there is no comment on the language in which the definitions are to be given. Examples such as the Peano axioms for the natural numbers and Hilbert's axioms for the real numbers suggest that second-order definitions should be allowed. As stated in s1930e, Zermelo is fully convinced that a set in the new sense "is precisely that which Cantor really meant by his well-known definition of "set', and it can be treated as a set everywhere and without contradiction in all purely mathematical considerations and deductions" (ibid., 5).