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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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Cited by 8 (2 self)
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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 inclusion vs ..."
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Cited by 5 (1 self)
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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 :
The Mathematical Infinite as a Matter of Method
, 2010
"... Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically ..."
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Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically discern two overlapping clusters of concepts: (1) wholeness, completeness, universality, absoluteness. (2) endlessness, boundlessness, indivisibility, continuousness. The first, the metaphysical infinite, I shall set aside. It is the second, the mathematical infinite, that I will address. Furthermore, I will address mathematical infinite by considering its historical emergence in set theory and how we are to take it in and out of mathematics. Insofar as physics and, more broadly, science deals with the mathematical infinite through mathematical language and techniques, my remarks should be subsuming and consequent. The main underlying point is that how the mathematical infinite is approached, assimilated, and applied in mathematics is not a matter of “ontological commitment”, of coming to terms with whatever that might mean, but rather of epistemological articulation, of coming to terms through knowledge. The mathematical infinite in mathematics is a matter of method. How we deal with the specific individual issues involving the infinite turns on the narrative we present about how it fits into methodological mathematical frameworks established and being established. The first section discusses the mathematical infinite in historical context, and the second, set theory and the emergence of the mathematical infinite. The third section discusses the infinite in and out of mathematics, and how it is to be taken. §1. The Infinite in Mathematics What role does the infinite play in modern mathematics? In modern mathematics, infinite sets abound both in the workings of proofs and as subject matter in statements, and so do universal statements, often of ∀ ∃ “for all there exists” form, which are indicative of direct engagement with the infinite. In many ways the role of the infinite is importantly “secondorder ” in the sense that Frege regarded number generally, in that the concepts of modern mathematics are understood as having infinite instances over a broad range. 1 But
Formalizing a Proof that e is Transcendental
, 2011
"... We describe a HOL Light formalization of Hermite’s proof that the base of the natural logarithm e is transcendental. This is the first time a proof of this fact has been formalized in a theorem prover. 1 ..."
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We describe a HOL Light formalization of Hermite’s proof that the base of the natural logarithm e is transcendental. This is the first time a proof of this fact has been formalized in a theorem prover. 1