Introductory note to s1930e

In this note found in the Nachlass, Zermelo describes his ideas in 1930 on the construction of a model of set theory and, with it, the consistency of set theory. As such, the note provides details on the first two topics that Zermelo had listed in the appendix, “Synopsis of further investigations currently in preparation”, of his report s1930d to the Notgemeinschaft der Deutschen Wissenschaft for the continuation of his fellowship into 1931. There are five carefully typed pages, the first four comprising a section “§ 1. The construction of the model” and dated 30 September 1930, and the last “On closed and open domains”. At least the first four pages were probably intended to be at the beginning of a publication.1 What is most notable about s1930e is that it shows Zermelo to be undoubtedly the first to provide an analysis of set theory and a motivation for its axioms based on what we now call the iterative conception. This is a way of thinking about the subject matter for set theory as stratified in layers, a set at a layer consisting of elements from former layers and the layers indexed with transfinite numbers. As sources for this one has referred to Zermelo 1930a and its adoption of the axiom of foundation and to Gödel’s work on L. In Gödel’s writings one sees the iterative conception as an underlying motivation in terms of an extension into higher types (cf. his publication 1947, 518ff and his lecture 1933 ). Specific articulations of the iterative conception as itself motivating the axioms of set theory appeared in the 1970s, e.g. Scott 1971, Shoenfield 1977, Wang 1974, and Boolos 1971. In Zermelo’s s1930e one now sees how the initial axiomatizer of set theory had already traveled down this path. Zermelo first sets out his set-theoretic model, and it is his cumulative hierarchy from 1930a based on a totality Q of urelements serving as a basis: