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The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusi ..."
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
Paradoxes in Göttingen
"... In 1903 Russell’s paradox came over the mathematical world with a double stroke. Bertrand Russell himself published it under the heading “The Contradiction” in chapter 10 of his Principles of Mathematics (Russell 1903). Almost at the same time Gottlob Frege (1848–1925) referred to Russell’s ..."
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In 1903 Russell’s paradox came over the mathematical world with a double stroke. Bertrand Russell himself published it under the heading “The Contradiction” in chapter 10 of his Principles of Mathematics (Russell 1903). Almost at the same time Gottlob Frege (1848–1925) referred to Russell’s
SET THEORY FROM CANTOR TO COHEN
"... Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun int ..."
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Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The
THE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR AKIHIRO KANAMORI
, 2002
"... For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks ..."
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For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of ’ {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary settheoretic concepts serves as a motif that reflects and illuminates larger and more significant developments in mathematical logic:
Zermelo: A well founded . . .
"... In his response to the letter from Paul Bernays (with congratulations on the occasion of his 70th anniversary) Zermelo wrote (October 1, 1941): Man wird eben immer einsamer, ist aber umso dankbarer für jedes freundliche Gedanken. [...] Wo mein Name noch genannt wird, geschieht es immer nur in Verbin ..."
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In his response to the letter from Paul Bernays (with congratulations on the occasion of his 70th anniversary) Zermelo wrote (October 1, 1941): Man wird eben immer einsamer, ist aber umso dankbarer für jedes freundliche Gedanken. [...] Wo mein Name noch genannt wird, geschieht es immer nur in Verbindung mit dem ‘Auswahlprinzip’, auf das ich niemals Prioritätsanspruche gestellt habe. [...] Dabei erinnere ich mich, daß schon bei der MathematikerTagung in Bad Elster mein Vortrag über SatzSysteme durch eine Intrige der von Hahn und Gödel vertretenen Wiener Schule von der Diskussion ausgeschlossen wurde, und habe seitdem die Lust verloren, über Grundlagen vorzutragen. So geht es augenscheinlich jedem, der keine ‘Schule ’ oder Klique hinter sich hat. Aber vielleicht kommt noch eine Zeit, wo auch meine Arbeiten wieder entdeckt und gelesen werden.1 Zermelo had in mind here his works listed below. They concern a foundational program formulated by him with special emphasis put on the infinitary (though always well founded) nature of mathematical proof. This idea is of course in sharp opposition to the (quite well established at that time) common uderstanding of the notion of finitary formal proof. Zermelo rejects what he himself calls Skolemism and the finitary prejudices: the views that set theory should be axiomatized in a first