Zermelo 1904

Abstract

On 24 September 1904, Zermelo sent a letter from “Münden in Hann.” to Hilbert. This letter contained a proof of the famous well-ordering theorem (WOT) from what was later called by Zermelo “the axiom of choice [Axiom der Auswahl]” (AC), a principle formulated for the first time in this letter. Shortly after, there appeared in volume 59 of the Mathematische Annalen a paper by Zermelo (no date is given for its receipt) of slightly less than three pages which sets out this proof. The title of the paper is “Beweis, daß jede Menge wohlgeordnet werden kann”, and it carried as a sub-title in brackets “Aus einem an Herrn Hilbert gerichteten Briefe”. The paper (Zermelo 1904 ) was destined to become one of the most celebrated in the history of modern mathematics, and the choice principle it contains one of modern mathematics’ most discussed and controversial principles. Unfortunately, Zermelo’s actual letter is not preserved among Hilbert’s correspondence or Zermelo’s papers, and its fate is not known. There do exist in Zermelo’s hand some fragments from a draft of this letter, but the bulk of the proof is missing. Thus, all that we really have is the published paper.