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Volumn 53, Issue 4, 2012, Pages 557-570

An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

Author keywords

Eudoxus; Hyperreals; Infinitesimals; Limit ultrapower; Universal hyperreal field

Indexed keywords


EID: 84869391881     PISSN: 00294527     EISSN: None     Source Type: Journal    
DOI: 10.1215/00294527-1722755     Document Type: Article
Times cited : (13)

References (42)
  • 2
    • 14644409847 scopus 로고
    • Nonstandard methods in stochastic analysis and mathematical physics
    • Academic Press, Orlando. Zbl pre06017936. MR 0859372
    • Albeverio, S., R. Høegh-Krohn, J. E. Fenstad, and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, vol. 22 of Pure and Applied Mathematics, Academic Press, Orlando, 1986. Zbl pre06017936. MR 0859372. 562
    • (1986) Pure and Applied Mathematics , vol.22 , pp. 562
    • Albeverio, S.1    Høegh-Krohn, R.2    Fenstad, J.E.3    Lindstrøm, T.4
  • 4
    • 0019717771 scopus 로고
    • Intermolecular forces of infinite range and the Boltzmann equation
    • Zbl 054776085 MR 0630119. 562
    • Arkeryd, L., "Intermolecular forces of infinite range and the Boltzmann equation," Archive for Rational Mechanics and Analysis, vol. 77 (1981), pp. 11-21. Zbl 0547.76085. MR 0630119. 562
    • (1981) Archive for Rational Mechanics and Analysis , vol.77 , pp. 11-21
    • Arkeryd, L.1
  • 5
    • 30344478815 scopus 로고    scopus 로고
    • Nonstandard analysis
    • Zbl 1128.03300. MR 2186838. 562
    • Arkeryd, L., "Nonstandard analysis," American Mathematical Monthly, vol. 112 (2005), pp. 926-28. Zbl 1128.03300. MR 2186838. 562
    • (2005) American Mathematical Monthly , vol.112 , pp. 926-928
    • Arkeryd, L.1
  • 6
    • 84880203668 scopus 로고    scopus 로고
    • An irrational construction of ℝ from ℤ
    • Theorem Proving in Higher Order Logics (Edinburgh, 2001), Springer, Berlin. Zbl 1005.68534. MR 1907599. 559, 567
    • Arthan, R., "An irrational construction of ℝ from ℤ," pp. 43-58 in Theorem Proving in Higher Order Logics (Edinburgh, 2001), vol. 2152 of Lecture Notes in Computer Science, Springer, Berlin, 2001. Zbl 1005.68534. MR 1907599. 559, 567
    • (2001) Lecture Notes in Computer Science , vol.2152 , pp. 43-58
    • Arthan, R.1
  • 7
    • 84880227216 scopus 로고    scopus 로고
    • preprint, arXiv:math/0405454v1 [math.HO], 558, 559, 567
    • Arthan, R., "The Eudoxus real numbers," preprint, arXiv:math/0405454v1 [math.HO] 557, 558, 559, 567
    • The Eudoxus Real Numbers , pp. 557
    • Arthan, R.1
  • 8
    • 84874545257 scopus 로고    scopus 로고
    • Ten misconceptions from the history of analysis and their debunking
    • published electronically March 22
    • Błasczcyk, P., M. Katz, and D. Sherry, "Ten misconceptions from the history of analysis and their debunking," Foundations of Science, published electronically March 22, 2012, http://dx.doi.org/10.1007/s10699-012- 9285-8 562
    • (2012) Foundations of Science , pp. 562
    • Błasczcyk, P.1    Katz, M.2    Sherry, D.3
  • 9
    • 84864312790 scopus 로고    scopus 로고
    • Who gave you the Cauchy-Weierstrass tale? the dual history of rigorous calculus
    • 562
    • Borovik, A., and M. Katz, "Who gave you the Cauchy-Weierstrass tale? The dual history of rigorous calculus," Foundations of Science, vol. 17 (2012), 245-76, http://dx.doi.org/10.1007/s10699-011-9235-X 562
    • (2012) Foundations of Science , vol.17 , pp. 245-276
    • Borovik, A.1    Katz, M.2
  • 10
    • 34547824184 scopus 로고    scopus 로고
    • A new look at E. G. Björling and the Cauchy sum theorem
    • Zbl 1151.01008. MR 2329097. 562
    • Bråting, K., "A new look at E. G. Björling and the Cauchy sum theorem," Archive for History of Exact Sciences, vol. 61 (2007), pp. 519-35. Zbl 1151.01008. MR 2329097. 562
    • (2007) Archive for History of Exact Sciences , vol.61 , pp. 519-535
    • Bråting, K.1
  • 11
    • 0002280322 scopus 로고
    • Model theory
    • 3rd edition, North-Holland, Amsterdam. Zbl 0697.03022. MR 1059055, 565, 566
    • Chang, C. C., and H. J. Keisler, Model Theory, 3rd edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1990. Zbl 0697.03022. MR 1059055. 562, 565, 566
    • (1990) Studies in Logic and the Foundations of Mathematics , vol.73 , pp. 562
    • Chang, C.C.1    Keisler, H.J.2
  • 12
    • 50949091104 scopus 로고    scopus 로고
    • 2nd corrected and expanded edition, Springer-Lehrbuch, Springer, Berlin. Zbl 1186.03065. MR 2841565
    • Deiser, O., Reelle Zahlen: Das klassische Kontinuum und die natürlichen Folgen, 2nd corrected and expanded edition, Springer-Lehrbuch, Springer, Berlin, 2008. Zbl 1186.03065. MR 2841565. 559
    • (2008) Reelle Zahlen: Das Klassische Kontinuum und Die Natürlichen Folgen , pp. 559
    • Deiser, O.1
  • 13
    • 28244451575 scopus 로고    scopus 로고
    • The rise of non-Archimedean mathematics and the roots of a misconception, I: The emergence of non-Archimedean systems of magnitudes
    • Zbl 1086.01024. MR 2206281. 559
    • Ehrlich, P., "The rise of non-Archimedean mathematics and the roots of a misconception, I: The emergence of non-Archimedean systems of magnitudes," Archive for History of Exact Sciences, vol. 60 (2006), pp. 1-121. Zbl 1086.01024. MR 2206281. 559
    • (2006) Archive for History of Exact Sciences , vol.60 , pp. 1-121
    • Ehrlich, P.1
  • 14
    • 84860122388 scopus 로고    scopus 로고
    • The absolute arithmetic continuum and the unification of all numbers great and small
    • Zbl pre06026116. 558, 564, 567
    • Ehrlich, P., "The absolute arithmetic continuum and the unification of all numbers great and small," Bulletin of Symbolic Logic, vol. 18 (2012), pp. 1-45. Zbl pre06026116. 558, 564, 567
    • (2012) Bulletin of Symbolic Logic , vol.18 , pp. 1-45
    • Ehrlich, P.1
  • 15
    • 77950283093 scopus 로고    scopus 로고
    • Nonstandard student conceptions about infinitesimals
    • 562
    • Ely, R., "Nonstandard student conceptions about infinitesimals, " Journal for Research in Mathematics Education, vol. 41 (2010), pp. 117-46. 562
    • (2010) Journal for Research in Mathematics Education , vol.41 , pp. 117-146
    • Ely, R.1
  • 17
    • 0013198293 scopus 로고    scopus 로고
    • Lectures on the hyperreals. An introduction to nonstandard analysis
    • Springer, New York. Zbl 0911.03032. MR 1643950
    • Goldblatt, R., Lectures on the Hyperreals. An Introduction to Nonstandard Analysis, vol. 188 of Graduate Texts in Mathematics, Springer, New York, 1998. Zbl 0911.03032. MR 1643950. 561
    • (1998) Graduate Texts in Mathematics , vol.188 , pp. 561
    • Goldblatt, R.1
  • 18
    • 22544438448 scopus 로고    scopus 로고
    • Describing the real numbers in terms of integers
    • Zbl 1098.16017. MR 2155112. 559
    • Grundhöfer, T., "Describing the real numbers in terms of integers," Archiv der Mathematik (Basel), vol. 85 (2005), pp. 79-81. Zbl 1098.16017. MR 2155112. 559
    • (2005) Archiv der Mathematik (Basel) , vol.85 , pp. 79-81
    • Grundhöfer, T.1
  • 19
    • 84935154381 scopus 로고
    • Rings of real-valued continuous functions, I
    • American Mathematical Society, Zbl 0032.28603. MR 0026239. 559, 561
    • Hewitt, E., "Rings of real-valued continuous functions, I," Transactions of the American Mathematical Society, vol. 64 (1948), pp. 45-99. Zbl 0032.28603. MR 0026239. 559, 561
    • (1948) Transactions of the , vol.64 , pp. 45-99
    • Hewitt, E.1
  • 20
    • 77953524058 scopus 로고
    • Correctness of the Euler method of decomposing the sine function into an infinite product
    • (in Russian); English translation in Russian Mathematical Surveys, vol. 49 (1988), 65-94. MR 0969566. 562
    • Kanovei, V. G., "Correctness of the Euler method of decomposing the sine function into an infinite product" (in Russian), Uspekhi Matematicheskikh Nauk, vol. 43 (1988), no. 4, pp. 57-81; English translation in Russian Mathematical Surveys, vol. 49 (1988), 65-94. MR 0969566. 562
    • (1988) Uspekhi Matematicheskikh Nauk , vol.43 , Issue.4 , pp. 57-81
    • Kanovei, V.G.1
  • 22
    • 77953701255 scopus 로고    scopus 로고
    • When is 1999 : : : Less than 1?
    • 562
    • Katz, K., and M. Katz, "When is 1999 : : : less than 1?," Montana Mathematics Enthusiast, vol. 7 (2010), pp. 3-30. 562
    • (2010) Montana Mathematics Enthusiast , vol.7 , pp. 3-30
    • Katz, K.1    Katz, M.2
  • 23
    • 77953696483 scopus 로고    scopus 로고
    • Zooming in on infinitesimal 1 - .9. in a post-triumvirate era
    • 562
    • Katz, K., and M. Katz, "Zooming in on infinitesimal 1 - .9. in a post-triumvirate era," Educational Studies in Mathematics, vol. 74 (2010), pp. 259-73. 562
    • (2010) Educational Studies in Mathematics , vol.74 , pp. 259-273
    • Katz, K.1    Katz, M.2
  • 24
    • 84857140393 scopus 로고    scopus 로고
    • Cauchy's continuum
    • MR 2884218. 562
    • Katz, K., and M. Katz, "Cauchy's continuum," Perspectives on Science, vol. 19 (2011), pp. 426-52. MR 2884218. 562
    • (2011) Perspectives on Science , vol.19 , pp. 426-452
    • Katz, K.1    Katz, M.2
  • 25
    • 84864303675 scopus 로고    scopus 로고
    • Meaning in classical mathematics: Is it at odds with intuitionism?
    • Katz, K., and M. Katz, "Meaning in classical mathematics: Is it at odds with intuitionism?" Intellectica, vol. 56 (2011), pp. 223-302.
    • (2011) Intellectica , vol.56 , pp. 223-302
    • Katz, K.1    Katz, M.2
  • 26
    • 84857142210 scopus 로고    scopus 로고
    • A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography
    • Katz, K., and M. Katz, "A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography," Foundations of Science, vol. 17 (2012), pp. 51-89.
    • (2012) Foundations of Science , vol.17 , pp. 51-89
    • Katz, K.1    Katz, M.2
  • 27
    • 84861774258 scopus 로고    scopus 로고
    • Stevin numbers and reality
    • 562
    • Katz, K., and M. Katz, "Stevin numbers and reality," Foundations of Science, vol. 17 (2012), pp. 109-23. 562
    • (2012) Foundations of Science , vol.17 , pp. 109-123
    • Katz, K.1    Katz, M.2
  • 29
    • 84876923301 scopus 로고    scopus 로고
    • Leibniz's infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond
    • published electronically April 20
    • Katz, M., and D. Sherry, "Leibniz's infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond," Erkenntnis, published electronically April 20, 2012, http://dx.doi.org/10.1007/s10670-012-9370-y 562
    • (2012) Erkenntnis , pp. 562
    • Katz, M.1    Sherry, D.2
  • 31
    • 84880216721 scopus 로고    scopus 로고
    • The tension between intuitive infinitesimals and formal mathematical analysis
    • Crossroads in the History of Mathematics and Mathematics Education edited by B Sriraman, Information Age Publishing, Charlotte, N.C. 562
    • Katz, M., and D. Tall, "The tension between intuitive infinitesimals and formal mathematical analysis," pp. 71-89 in Crossroads in the History of Mathematics and Mathematics Education, edited by B. Sriraman, vol. 12 of The Montana Mathematics Enthusiast Monographs in Mathematics Education, Information Age Publishing, Charlotte, N.C., 2012. 562
    • (2012) The Montana Mathematics Enthusiast Monographs in Mathematics Education , vol.12 , pp. 71-89
    • Katz, M.1    Tall, D.2
  • 32
  • 33
    • 0038965663 scopus 로고
    • Ultrafilters and independent sets
    • Zbl 0263.02033. MR 0314619. 566
    • Kunen, K., "Ultrafilters and independent sets," Transactions of the American Mathematical Society, vol. 172 (1972), pp. 299-306. Zbl 0263.02033. MR 0314619. 566
    • (1972) Transactions of the American Mathematical Society , vol.172 , pp. 299-306
    • Kunen, K.1
  • 34
    • 0003304491 scopus 로고
    • Set theory: An introduction to independence proofs
    • North-Holland, Amsterdam. Zbl 0443.03021. MR 0597342
    • Kunen, K., Set Theory: An Introduction to Independence Proofs, vol. 102 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1980. Zbl 0443.03021. MR 0597342. 567
    • (1980) Studies in Logic and the Foundations of Mathematics , vol.102 , pp. 567
    • Kunen, K.1
  • 36
    • 84874532813 scopus 로고    scopus 로고
    • Operational semantics for timed systems
    • 562
    • Rust, H., "Operational semantics for timed systems," Lecture Notes in Computer Science, vol. 3456 (2005), pp. 23-29. 562
    • (2005) Lecture Notes in Computer Science , vol.3456 , pp. 23-29
    • Rust, H.1
  • 37
    • 0001844026 scopus 로고
    • Eine erweiterung der infinitesimalrechnung
    • Zbl 0082.04203. MR 0095906. 560
    • Schmieden, C., and D. Laugwitz, "Eine Erweiterung der Infinitesimalrechnung," Mathematische Zeitschrift, vol. 69 (1958), pp. 1-39. Zbl 0082.04203. MR 0095906. 560
    • (1958) Mathematische Zeitschrift , vol.69 , pp. 1-39
    • Schmieden, C.1    Laugwitz, D.2
  • 38
    • 65749311389 scopus 로고
    • A topics course in mathematics
    • Zbl 0623.01029. 567
    • Shenitzer, A., "A topics course in mathematics," The Mathematical Intelligencer, vol. 9 (1987), pp. 44-52. Zbl 0623.01029. 567
    • (1987) The Mathematical Intelligencer , vol.9 , pp. 44-52
    • Shenitzer, A.1
  • 39
    • 0040534583 scopus 로고
    • Über die Nicht-charakterisierbarkeit der zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen
    • 560
    • Skolem, T., "Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen," Fundamenta Mathamaticae, vol. 23 (1934), pp. 150-61. 560
    • (1934) Fundamenta Mathamaticae , vol.23 , pp. 150-161
    • Skolem, T.1
  • 41
    • 84880203454 scopus 로고
    • Leopold kronecker
    • 557
    • Weber, M. "Leopold Kronecker," Mathematische Annalen, vol. 43 (1893), pp. 1-25. 557
    • (1893) Mathematische Annalen , vol.43 , pp. 1-25
    • Weber, M.1
  • 42
    • 79959277676 scopus 로고
    • Book review: The mathematical career of Pierre de Fermat
    • MR 1566693. 567
    • Weil, A., "Book Review: The mathematical career of Pierre de Fermat," Bulletin of the American Mathematical Society, vol. 79 (1973), pp. 1138-49. MR 1566693. 567
    • (1973) Bulletin of the American Mathematical Society , vol.79 , pp. 1138-1149
    • Weil, A.1


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