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Constructive ReasoningIn B. Van Rootselaar & J. F. Staal (eds.), Logic, Methodology and Philosophy of Science III, North-holland. pp. 185-99. 1968.
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14J. P. Mayberry. The foundations of mathematics in the theory of sets. Encyclopedia of mathematics and its applications, vol. 82. Cambridge University Press, Cambridge 2000, New York 2001, etc., xx + 424 pp (review)Bulletin of Symbolic Logic 8 (3): 424-426. 2002.
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Frege versus Cantor and Dedekind: On the Concept of NumberIn Matthias Schirn (ed.), Frege: importance and legacy, Walter De Gruyter. pp. 70-113. 1996.
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1What Hilbert and Bernays Meant by "Finitism"In Gabriele Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium, De Gruyter. pp. 249-261. 2019.
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69Noesis: Plato on exact scienceIn David B. Malament (ed.), Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics, Open Court. pp. 11--31. 2002.
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25The Hilton New York Hotel New York, NY December 27–29, 2005Bulletin of Symbolic Logic 12 (3). 2006.
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150Gödel's reformulation of Gentzen's first consistency proof for arithmetic: The no-counterexample interpretationBulletin of Symbolic Logic 11 (2): 225-238. 2005.The last section of “Lecture at Zilsel’s” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Godel's reformulation, and discuss the relation between the two proofs.
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23Extensional Equality in the Classical Theory of TypesVienna Circle Institute Yearbook 3 219-234. 1995.The classical theory of types in question is essentially the theory of Martin-Löf [1] but with the law of double negation elimination. I am ultimately interested in the theory of types as a framework for the foundations of mathematics and, for this purpose, we need to consider extensions of the theory obtained by adding ‘well-ordered types,’ for example the type N of the finite ordinals; but the unextended theory will suffice to illustrate the treatment of extensional equality
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23Review: Steven Orey, On $omega$-Consistency and Related Properties (review)Journal of Symbolic Logic 23 (1): 40-41. 1958.
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17A Nonconstructive Proof of Gentzen's Hauptsatz for Second Order Predicate LogicJournal of Symbolic Logic 33 (2): 289-290. 1968.
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34Plato's Second Best MethodReview of Metaphysics 39 (3). 1986.AT PHAEDO 96A-C Plato portrays Socrates as describing his past study of "the kind of wisdom known as περὶ φυσέως ἱστορία." At 96c-97b, Socrates says that this study led him to realize that he had an inadequate understanding of certain basic concepts which it involved. In consequence, he says at 97b, he abandoned this method and turned to a method of his own. But at this point in the dialogue, instead of proceeding immediately to describe his method, Plato has him interjecting a complaint concern…Read more
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32Kurt Godel. Collected Works. Volume IV: Selected Correspondence AG; Volume V: Selected Correspondence HZPhilosophia Mathematica 14 (1): 76. 2006.
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To appear in the Proceedings of Logic Colloquium 2006. (28 pages).
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61Curtis Franks The Autonomy of Mathematical Knowledge: Hilbert's Program RevisitedHistory and Philosophy of Logic 32 (2). 2011.History and Philosophy of Logic, Volume 32, Issue 2, Page 177-183, May 2011
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12Review: H. G. Rice, On Completely Recursively Enumerable Classes and Their Key Arrays (review)Journal of Symbolic Logic 23 (1): 48-48. 1958.
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30The Palmer House Hilton Hotel, Chicago, Illinois April 19–21, 2007Bulletin of Symbolic Logic 13 (4). 2007.
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129Intensional interpretations of functionals of finite type IJournal of Symbolic Logic 32 (2): 198-212. 1967.
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72The law of excluded middle and the axiom of choiceIn Alexander George (ed.), Mathematics and Mind, Oxford University Press. pp. 45--70. 1994.
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78Some recent essays in the history of the philosophy of mathematics: A critical review (review)Synthese 96 (2). 1993.
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108Proof-theoretic Semantics for Classical MathematicsSynthese 148 (3): 603-622. 2006.We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equalit…Read more
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