INTRODUCING BOTH "THE AXIOM OF CHOICE" AND "THE WELL-ORDERING THEOREM"

ZERMELO, ERNST FRIEDRICH FERDINAND.

Beweis, dass jede Menge wohlgeordnet werden kann. (Aus einem an Herrn Hilbert gerichteten Briefe).

Leipzig, B.G. Teubner, 1904. 8vo. Bound in half cloth with five raised bands and gilt lettering to spine. In "Mathematische Annalen. Begründet 1868 durch von A. Clebsch und C. Neumann. 59. Band." Small bar-code pasted on to top left corner of front wrapper. Two library labels pasted on to pasted down front free end-paper and a small stamp to verso of title page. Pp. 514-16.

First appearence of this fundamental paper in metamathematics and mathematical logic in which he introduced both "The Axiom of Choice" and "his sensational proof of the well-ordering theorem" (DSB).
By this paper Zermelo contributed decisively to the development of set-theory. Zermelo took up the problem, left over by Cantor, of what to do about the comparison of sets that are not well-ordered. "In 1904 (the paper offered) he proved....that every set can be well-ordered. To make the proof he had to use what is now known as the axiom of choice (Zermelo's axiom), which states that given any collection of nonempty, disjoined sets, it is possible to choose one member from each set and so make up a new set. The axiom of choce, the well-ordering theorem, and the fact that any two sets may be compared as to size, are equivalent principles."(Morris Kline). A controversy arose around "The axiom of Choice", from Bertrand Russell, Tarski, Frege, Hilbert, Brouwer and others, mainly, and of course importent, over how to interpret the words "choose" and "exists".

The issue also contain the following papers of interest:
David Hilbert. Über das dirichletsche prinzip. Pp. 161-186.
Lie, Sophus. Drei Kapitel aus dem unvollendeten zweiten Bande der Geometrie der Berührungstransformationen. Pp. 193-313.
Schoenflies, A. Beitrage zur Theorie der Punktmengen II, Pp. 129-160.

Order-nr.: 44952