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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 :
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
A Historical Account of SetTheoretic Antinomies Caused by the Axiom of Abstraction
"... This paper will be concerned with a historical description of the prominent paradoxes that resulted from the use of Cantor's denition and the Axiom of Abstraction near the beginning of the twentieth century. The distinction between `paradox' and `contradiction' will be used as in [6]. ..."
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This paper will be concerned with a historical description of the prominent paradoxes that resulted from the use of Cantor's denition and the Axiom of Abstraction near the beginning of the twentieth century. The distinction between `paradox' and `contradiction' will be used as in [6]. Thus, a contradiction will occur when a statement and its negation can both be proved true, but a paradox will be dened as \an argument which ends in a contradiction although all of its premises and modes of reasoning are prima facie acceptable" ([6], p. 321), with the further stipulation that \the one who discovers it give up a premise or mode of reasoning that he has previously accepted as correct" ([6], p. 321). In this case, the Axiom of Abstraction will be the abandoned premise, given up for either Zermelo's Axiom of Separation or Fraenkel's Axiom Schema of Replacement, which implies Zermelo's Axiom
TITLE: A History ofthe Theory of Types with Special Reference to Developments After the Second Edition of Principia Mathematica
"... This thesis traces the development ofthe theory of types from its origins in the early twentieth century through its various forms until the mid 1950's. Special attention is paid to the reception of this theory after the publication of the second edition of Whitehead and Russell's Principi ..."
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This thesis traces the development ofthe theory of types from its origins in the early twentieth century through its various forms until the mid 1950's. Special attention is paid to the reception of this theory after the publication of the second edition of Whitehead and Russell's Principia Mathematica. We examine how the theory of types declined in influence over four decades. From being in the 1920s the dominant form of mathematical logic, by 1956 this theory had been abandoned as a foundation for mathematics. The use and modification ofthe theory by logicians such as Ramsey, Carnap, ChurctU Quine, Gddel, and Tarski is given particular attention. Finally, the view of the theory of types as a manysorted firstorder theory in the 1950's is discussed. It was the simple theory, as opposed to the ramified theory of types that was used almost exclusively during the years following the second edition of Principia. However, it is shown in this thesis that in the 1950's a revival of the ramified theory of types occurred. This revival of ramifiedtype theories coincided with the consideration of cumulative type hierarchies. This is most evident in the work of Hao Wang and John Myhill. The consideration of cumulative tlpehierarchies altered the form of the theory of types in a substantial way. The theory was altered even more drastically by being changed from a manysorted theory into a onesorted theory. This final "standardization" of the theory of types in the mid 1950's made it not much different from firstorder ZermeloFraenkel settheory. The theory oftypes, whose developments are traced in this thesis, therefore lost its prominence as the foundation for mathematics and logic. ur
FROM MOORE TO PEANO TO WATSON The Mathematical Roots of Russell’s Naturalism and Behaviorism INTRODUCTION: SOME ISSUES REGARDING RUSSELL’S PHILOSOPHICAL DEVELOPMENT
"... Russell’s philosophical development is marked by a number of key shifts in his outlook that he vividly describes in his retrospective writings. Among these are his “becom[ing] a Hegelian ” in 1894 (1944a, 10; ..."
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Russell’s philosophical development is marked by a number of key shifts in his outlook that he vividly describes in his retrospective writings. Among these are his “becom[ing] a Hegelian ” in 1894 (1944a, 10;