The origins of Zermelo's axiomatization of set theory

Abstract

What gave rise to Ernst Zermelo's axiomatization of set theory in 1908? According to the usual interpretation, Zermelo was motivated by the set-theoretic paradoxes. This paper argues that Zermelo was primarily motivated, not by the paradoxes, but by the controversy surrounding his 1904 proof that every set can be well-ordered, and especially by a desire to preserve his Axiom of Choice from its numerous critics. Here Zermelo's concern for the foundations of mathematics diverged from Bertrand Russell's on the one hand and from Felix Hausdorff's on the other.