Introductory note to 1930a

Zermelo in his remarkable 1930a offered his final axiomatization of set theory as well as a striking, synthetic view of a procession of natural models that would have a modern resonance. Appearing only six articles after Skolem 1930 in Fundamenta mathematicae, Zermelo’s article seemed strategically placed as a response, an aspect that we will discuss below, but its dramatically new picture of set theory reflects gained experience and suggests the germination of ideas over a prolonged period. The subtitle, “New investigations in the foundations of set theory”, evidently recalls his axiomatization article 1908b, differing only in the “New” from the title of that article. The new article is a tour de force which sets out principles that would be adopted in the further development of set theory and draws attention to the cumulative hierarchy picture, dialectically enriched by initial segments serving as natural models. In Section 1, Zermelo formulates his axiom system, the “constitutive axioms” of “general set theory”, and though the presentation is opaque largely because of a second-order lens, the thrust of ZFC is there. Indeed, Zermelo used the term “Zermelo-Fraenkel” to indicate the result of adding the replacement axiom (which subsumes the separation axiom) to his 1908b axioms.1 Zermelo actually proceeded with his “ZF 0-system”, the result of deleting the axiom of infinity as not being part of general set theory; assuming the axiom of choice as an implicit underlying “general logical principle”; and newly adjoining the axiom of foundation—the term having its source in this article. Concerning the replacement axiom, Zermelo in a letter of 9 May 1921 to Abraham Fraenkel had formulated a version and aired, though with skepticism, the possibility of adopting it as a new axiom.2 Fraenkel advocated that adoption in a paper 1922b, completed on 1 July 1921, through which he would become associated with the axiom. It was however the work of von Neumann 1928d, done in the early 1920s, that made evident the importance of adopting replacement, for the formalization of transfinite recursion and the existence of sets defined therewith. Concerning the foundation axiom, in modern notation the axiom, as is well-known, entails that the universe V of sets is stratified into cumulative ranks V