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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.
Bachelier and his Times: A Conversation with Bernard Bru
, 2001
"... Louis Bachelier defended his thesis “Theory of Speculation” in 1900. He used Brownian motion as a model for stock exchange performance. This conversation with Bernard Bru illustrates the scientific climate of his times and the conditions under which Bachelier made his discoveries. It indicates that ..."
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Louis Bachelier defended his thesis “Theory of Speculation” in 1900. He used Brownian motion as a model for stock exchange performance. This conversation with Bernard Bru illustrates the scientific climate of his times and the conditions under which Bachelier made his discoveries. It indicates that Bachelier was indeed the right person at the right time. He was involved with the Paris stock exchange, was selftaught but also took courses in probability and on the theory of heat. Not beinga part of the “scientific establishment,” he had the opportunity to develop an area that was not of interest to the mathematicians of the period. He was the first to apply the trajectories of Brownian motion, and his theories prefigure modern mathematical finance. What follows is an edited and expanded version of the original conversation with Bernard Bru.
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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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 Halting Probability Omega: Irreducible Complexity in Pure Mathematics ∗
"... Some Gödel centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. Introduction: What is mathematics? It is a pleasure for me to be here today giving this ta ..."
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Some Gödel centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. Introduction: What is mathematics? It is a pleasure for me to be here today giving this talk in a lecture series in honor of Frederigo Enriques. Enriques was a great believer in mathematical intuition, and disdained formal proofs. The work of Gödel, Turing and myself that I will review goes some way to justifying Enriques’s belief in intuition.
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providi ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
Tmartingales, sizebiasing and tree polymer cascades
"... Directed lattice polymers on the d + 1 dimensional integer lattice are modeled by (random) distributions of graphs of polygonal paths in N × Z d for which the horizontal coordinate serves to direct the path as a selfavoiding chain of connected monomers. ..."
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Directed lattice polymers on the d + 1 dimensional integer lattice are modeled by (random) distributions of graphs of polygonal paths in N × Z d for which the horizontal coordinate serves to direct the path as a selfavoiding chain of connected monomers.
Computability Theory, Algorithmic Randomness and Turing’s Anticipation
"... Abstract. This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript. 1 ..."
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Abstract. This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript. 1