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170The mathematical development of set theory from Cantor to CohenBulletin of Symbolic Logic 2 (1): 1-71. 1996.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 cast in set-theoretic terms 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 crise…Read more
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158Handbook of the history of logic (edited book)Elsevier. 2004.Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, includ…Read more
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123The mathematical import of zermelo's well-ordering theoremBulletin of Symbolic Logic 3 (3): 281-311. 1997.Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In 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…Read more
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106Zermelo and set theoryBulletin of Symbolic Logic 10 (4): 487-553. 2004.Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framewo…Read more
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102Cohen and set theoryBulletin of Symbolic Logic 14 (3): 351-378. 2008.We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing
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88The empty set, the Singleton, and the ordered pairBulletin of Symbolic Logic 9 (3): 273-298. 2003.For the modern set theorist the empty set Ø, 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 locks 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 Ch…Read more
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81In praise of replacementBulletin of Symbolic Logic 18 (1): 46-90. 2012.This article serves to present a large mathematical perspective and historical basis for the Axiom of Replacement as well as to affirm its importance as a central axiom of modern set theory.
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69Gödel and set theoryBulletin of Symbolic Logic 13 (2): 153-188. 2007.Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic…Read more
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53Bernays and set theoryBulletin of Symbolic Logic 15 (1): 43-69. 2009.We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles
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49Atlanta Marriott Marquis, Atlanta, Georgia January 7–8, 2005Bulletin of Symbolic Logic 11 (3). 2005.
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46Levy and set theoryAnnals of Pure and Applied Logic 140 (1): 233-252. 2006.Azriel Levy did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of…Read more
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362004–05 Winter Meeting of the Association for Symbolic Logic (review)Bulletin of Symbolic Logic 11 (3): 454-460. 2005.
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29Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics (review)Bulletin of Symbolic Logic 7 (2): 277-278. 2001.
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28The compleat 0†Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (2): 133-141. 1990.
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27Mathematical Knowledge : Motley and Complexity of ProofAnnals of the Japan Association for Philosophy of Science 21 21-35. 2013.
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26Erdős and set theoryBulletin of Symbolic Logic 20 (4). 2014.Paul Erdős was a mathematicianpar excellencewhose results and initiatives have had a large impact and made a strong imprint on the doing of and thinking about mathematics. A mathematician of alacrity, detail, and collaboration, Erdős in his six decades of work moved and thought quickly, entertained increasingly many parameters, and wrote over 1500 articles, the majority with others. Hismodus operandiwas to drive mathematics through cycles of problem, proof, and conjecture, ceaselessly progressin…Read more
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24On Gödel incompleteness and finite combinatoricsAnnals of Pure and Applied Logic 33 (C): 23-41. 1987.
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24Shelah Saharon. Cardinal arithmetic. Oxford logic guides, no. 29. Clarendon Press, Oxford University Press, Oxford and New York1994, xxxi + 481 pp (review)Journal of Symbolic Logic 62 (3): 1035-1039. 1997.
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23The Mathematical Infinite as a Matter of MethodAnnals of the Japan Association for Philosophy of Science 20 3-15. 2012.
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222000 Annual Meeting of the Association for Symbolic LogicBulletin of Symbolic Logic 6 (3): 361-396. 2000.
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22Review: Jon Barwise, H. J. Keisler, K. Kunen, Y. N. Moschovakis, A. S. Troelstra, Handbook of Mathematical Logic; J. R. Shoenfield, B.1. Axioms of Set Theory (review)Journal of Symbolic Logic 49 (3): 971-975. 1984.
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Boston UniversityRegular Faculty
Boston, Massachusetts, United States of America
Areas of Interest
Logic and Philosophy of Logic |
Philosophy of Mathematics |