Zermelo 1930a

pp. 390-430

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

Abstract

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.