Results 1 to 10 of about 1,578 (68)

Cauchy, infinitesimals and ghosts of departed quantifiers [PDF]

, 2017
Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters.
Bair, Jacques, Blaszczyk, Piotr, Ely, Robert, Henry, Valerie, Kanovei, Vladimir, Katz, Karin U., Katz, Mikhail G., Kudryk, Taras, Kutateladze, Semen S., McGaffey, Thomas, Mormann, Thomas, Schaps, David M., Sherry, David   +12 more
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Stevin numbers and reality [PDF]

, 2012
We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.Comment: 22 pages, 4 figures.
A. H. Lightstone, A. L. Cauchy, A. Malet, A. Robinson, A. Tarski, A.-L. Cauchy, B. L. Waerden van der, C.S. Peirce, C.S. Peirce, D. Fearnley-Sander, D. H. Fowler, D. Laugwitz, D. Laugwitz, D. Laugwitz, D. Laugwitz, D. Sherry, D. Tall, D. Tall, E. Hewitt, G. Hellman, G. Lakoff, G. Schubring, H. Freudenthal, H. Sinaceur, J. Bair, J. Burgess, J. Dauben, J. Dauben, J. Havenel, J. L. Bell, K. Bråting, K. Katz, Karin Usadi Katz, L. Arkeryd, L. Arkeryd, L. Magnani, M. Moore, M. Moore, Mikhail G. Katz, N. Cutland, P. Ehrlich, P. Roquette, R. Dossena, R. Ely, R. Goldblatt, S. Gandz, Th. Skolem   +46 more
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The prospects for mathematical logic in the twenty-first century [PDF]

, 2001
The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed ...
Alexander S. Kechris, Anand Pillay, Richard A, Richard A. Shore, Samuel R. Buss   +4 more
core   +3 more sources

A non-standard analysis of a cultural icon: The case of Paul Halmos [PDF]

, 2016
We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism.
Blaszczyk, Piotr, Borovik, Alexandre, Kanovei, Vladimir, Katz, Mikhail G., Kudryk, Taras, Kutateladze, Semen S., Sherry, David   +6 more
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Infinity [PDF]

, 2007
This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2.
Fletcher, P
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Approaches to analysis with infinitesimals following Robinson, Nelson, and others [PDF]

, 2017
This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended ...
Fletcher, P, Hrbacek, K, Kanovei, V, Katz, MG, Lobry, C, Sanders, S   +5 more
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Ten Misconceptions from the History of Analysis and Their Debunking [PDF]

, 2012
The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems.
A. Connes, A. Fraenkel, A. H. Lightstone, A. L. Cauchy, A. L. Cauchy, A. Malet, A. Robinson, A. Tarski, A. Weil, A. Weil, A.L. Cauchy, B. L. Waerden van der, B. Pourciau, B. Russell, C. Baltus, C. Boyer, D. E. Smith, D. Fearnley-Sander, D. H. Fowler, D. Laugwitz, D. Laugwitz, D. Laugwitz, D. Laugwitz, D. Sepkoski, D. Sherry, D. Sherry, D. Tall, D. Tall, David Sherry, E. Hewitt, E. Knobloch, E. Madison, E. Nelson, E.T. Bell, F. Faltin, F. W. Lawvere, G. Fisher, G. Leibniz, G. Schubring, H. Freudenthal, H. J. Keisler, H. J. M. Bos, H. Putnam, H. Rust, H. Sinaceur, Huygens, I. Grattan-Guinness, I. Kleiner, I. Lakatos, J. Crossley, J. Dauben, J. Grabiner, J. Gray, J. L. Bell, J. Lützen, J. Naets, K. Andersen, K. Bråting, K. Hrbacek, K. Hrbacek, K. Hrbáček, K. Katz, L. Arkeryd, L. Arkeryd, L. Magnani, M. Dehn, M. J. Barany, Mikhail G. Katz, N. Cutland, P. Ehrlich, P. Ehrlich, P. Giordano, P. Giordano, P. Mancosu, P. Mancosu, P. Roquette, P. Strømholm, Piotr Błaszczyk, R. Anderson, R. B. McClenon, R. Dedekind, R. Dedekind, R. Dossena, R. Ely, R. Lutz, S. Albeverio, S. H. Levy, S. Smale, T. Koetsier, T. Mormann, T. Tao, Th. Skolem, V. Benci, V. J. Katz, W. Felscher, W. Felscher   +95 more
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Incompleteness via paradox and completeness [PDF]

, 2020
This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and ...
Bernays, Bernays, Bernays, Bernays, Bernays, Boolos, Borel, Church, Cieśliński, Cohen, Dummett, Enayat, Ewald, Feferman, Feferman, Fitch, Gödel, Hilbert, Hilbert, Hilbert, Hilbert, Hilbert, Hilbert, Hilbert, Hilbert, Hilbert, Hájek, Isaacson, Kaye, Kleene, Kreisel, Kreisel, Kripke, Kritchman, Lebesgue, Lusin, Mancosu, McGee, Mendelson, Mostowski, Müller, Nelson, Nelson, Putnam, Quine, Read, Rogers, Russell, Smorynski, Tarski, Tarski, Tarski, van Heijenoort, Visser, Von Neumann, Vopĕnka, Vopĕnka, WALTER DEAN, Wang, Wang, Weyl, Weyl   +61 more
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When is .999... less than 1? [PDF]

, 2010
We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities?
Katz, Karin Usadi, Katz, Mikhail G.
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From Nonstandard Analysis to various flavours of Computability Theory

, 2017
As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA).
A Connes, A Connes, A Robinson, AS Troelstra, B Berg van den, E Bishop, E Nelson, HJ Keisler, HJ Keisler, MG Katz, U Kohlenbach, V Kanovei   +11 more
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