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Beweis, dass jede Menge wohlgeordnet werden kann. (Aus einem an Herrn Hilbert gerichteten Briefe). - [INTRODUCING BOTH ""THE AXIOM OF CHOICE"" AND ""THE WELL-ORDERING THEOREM""]
Leipzig, B.G. Teubner, 1904. With orig. printed wrappers (no backstrip) to 4. Heft, 59. Bd. of ""Mathematische Annalen"". The issue: pp. 449-572. Zermelo's paper: pp. 514-516.
First appearence of this fundamental paper in metamathematics and mathematical logic. 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"".
Neuer Beweis für die Möglichkeit einer Wohlordnung (+) Untersuchungen übe die Grundlagen der Mengenlehre I. - [THE BASIS OF MODERN SET-THEORY]
Leipzig, B.G. Teubner, 1908. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 65., 1908. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of verso of title page. Very fine and clean. Pp. 107-128"" Pp. 261-181. [Entire volume: Pp. IV, 575, (1).]
First publication of these landmark paper's which ""has proved of tremendous importance for the development of mathematics"", (DSB) as they constitute the first formulation of the axiomatization of ""Set-theory"".In the first paper offered here (1907/1908) he gave a new proof of the well-ordering theorem (the first in his paper from 1904) as a reply to the attacks on his 1904-paper and he shows then that a limited number of specific principles, including a version of choice, is sufficient to deduce the well-ordering theorem. The two papers here offered, are closely connected, and in the second paper he formulated the FIRST AXIOMATIZATION OF SET-THEORY, avoiding the known antinomies, and thus was to become the basis of moder set.theory.The axioms here set up for Cantor's theory of sets, in all 7 axioms, was to ""save the theory from paradoxes. regarding this theory (Russel's theory of types), still as the most fundamental part of mathematics, he suggests that it should be rebuild by the laying down of principles which are sufficient to support the generally accepted doctrine but so chosen that they do not give rise to contradictions. He admits that he cannot prove the consistency of his axioms, but he claims that he has at least excluded antinomies discovered in recent years. The essential feature of his method is that he no longer talks of sets with the freedom of Cantor, but admits in his theory only those sets whose existence is guranteed by his axioms."" (W. a. M. Kneale in ""The Development of Logic).""The historian Gregory Moore has argued that it was not the discovery of the paradoxes, nor Russell's proposals (in his 1906) of three ways to avoid them, that impelled Zermelo to axiomatize set theory, but rather his determination to secure the acceptance of his well-ordering theorem. In support of that contention he points out that Zermelo had independently discovered ""Russell's"" paradox himself but had not found it troubling enough to publish, and he remarks that in his paper (i.e. the first paper offered) Zermelo employed the paradoxes ""merely as a club with which to bludgeon (his) critics"" (Moore 1982, pp. 158-159).
Beweis, dass jede Menge wohlgeordnet werden kann. (Aus einem an Herrn Hilbert gerichteten Briefe). - [INTRODUCING BOTH ""THE AXIOM OF CHOICE"" AND ""THE WELL-ORDERING THEOREM""]
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.
