×

An integer construction of infinitesimals: toward a theory of eudoxus hyperreals. (English) Zbl 1266.03074

Say a function \(f:\mathbb Z\to\mathbb Z\) is an almost homomorphism if there is \(C\in \mathbb N^{>0}\) such that, for all \(k,l\in \mathbb Z\), one has \(|f(k+l)-f(k)-f(l)|<C\). We identify two almost homomorphisms if they differ by a bounded function on \(\mathbb Z\). The set of equivalence classes can be equipped with the structure of an ordered field in a natural way, called the ordered field of Eudoxus reals \(\mathbb E\). Moreover, the map which sends the real number \(\alpha\) to the equivalence class of the almost homomorphism sending \(k\) to the integer part of \(k\alpha\) is an isomorphism of complete ordered fields. It is in this way that the Eudoxus reals can be seen as a construction of the real numbers directly from the integers, thereby “skipping the rationals”.
In this article, the above construction is combined with the ultrapower (or limit ultrapower) construction to construct the hyperreal fields of nonstandard analysis directly from the integers.

MSC:

03H05 Nonstandard models in mathematics
03C20 Ultraproducts and related constructions
26E35 Nonstandard analysis
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] A’Campo, N., “A natural construction for the real numbers,” preprint, [math.GN]
[2] Albeverio, S., R. Høegh-Krohn, J. E. Fenstad, and T. Lindstrø m, Nonstandard Methods in Stochastic Analysis and Mathematical Physics , vol. 22 of Pure and Applied Mathematics , Academic Press, Orlando, 1986. · Zbl 0605.60005
[3] Anderson, R. M., “Infinitesimal methods in mathematical economics,” preprint, 2008.
[4] 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 · doi:10.1007/BF00280403
[5] Arkeryd, L., “Nonstandard analysis,” American Mathematical Monthly , vol. 112 (2005), pp. 926-28. · Zbl 1128.03300 · doi:10.2307/30037635
[6] Arthan, R., “An irrational construction of \(\mathbb{R}\) from \(\mathbb{Z}\),” 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 · doi:10.1007/3-540-44755-5_5
[7] Arthan, R., “The Eudoxus real numbers,” preprint, [math.HO]
[8] 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, · Zbl 1291.01018
[9] 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, · Zbl 1279.01017
[10] 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 · doi:10.1007/s00407-007-0005-7
[11] 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
[12] Deiser, O., Reelle Zahlen: Das klassische Kontinuum und die natürlichen Folgen , 2nd corrected and expanded edition, Springer-Lehrbuch , Springer, Berlin, 2008.
[13] 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 · doi:10.1007/s00407-005-0102-4
[14] 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 1242.03065 · doi:10.2178/bsl/1327328438
[15] Ely, R., “Nonstandard student conceptions about infinitesimals,” Journal for Research in Mathematics Education , vol. 41 (2010), pp. 117-46.
[16] Giordano, P., and M. Katz, “Two ways of obtaining infinitesimals by refining Cantor’s completion of the reals,” preprint, [math.LO] 1109.3553v1
[17] 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
[18] Grundhöfer, T., “Describing the real numbers in terms of integers,” Archiv der Mathematik (Basel) , vol. 85 (2005), pp. 79-81. · Zbl 1098.16017 · doi:10.1007/s00013-005-1413-z
[19] Hewitt, E., “Rings of real-valued continuous functions, I,” Transactions of the American Mathematical Society , vol. 64 (1948), pp. 45-99. · Zbl 0032.28603 · doi:10.2307/1990558
[20] 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.
[21] Kanovei, V. G., and M. Reeken, Nonstandard Analysis, Axiomatically , Springer Monographs in Mathematics, Springer, Berlin, 2004. · Zbl 1058.03002
[22] Katz, K., and M. Katz, “When is \(.999\ldots\) less than \(1\)?,” Montana Mathematics Enthusiast , vol. 7 (2010), pp. 3-30.
[23] 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.
[24] Katz, K., and M. Katz, “Cauchy’s continuum,” Perspectives on Science , vol. 19 (2011), pp. 426-52. · Zbl 1292.01028 · doi:10.1162/POSC_a_00047
[25] Katz, K., and M. Katz, “Meaning in classical mathematics: Is it at odds with intuitionism?” Intellectica , vol. 56 (2011), pp. 223-302.
[26] 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. · Zbl 1283.03006 · doi:10.1007/s10699-011-9223-1
[27] Katz, K., and M. Katz, “Stevin numbers and reality,” Foundations of Science , vol. 17 (2012), pp. 109-23. · Zbl 1275.01016
[28] Katz, M., and E. Leichtnam, “Commuting and non-commuting infinitesimals,” to appear in American Mathematical Monthly . · Zbl 1280.26047
[29] 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, · Zbl 1303.01012
[30] Katz, M., and D. Sherry, “Leibniz’s laws of continuity and homogeneity,” Notices of the American Mathematical Society , vol. 59 (2012), no. 11. · Zbl 1284.03064
[31] 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.
[32] Keisler, H. J., “Limit ultrapowers,” Transactions of the American Mathematical Society , vol. 107 (1963), pp. 382-408. · Zbl 0122.24504 · doi:10.2307/1993808
[33] Kunen, K., “Ultrafilters and independent sets,” Transactions of the American Mathematical Society , vol. 172 (1972), pp. 299-306. · Zbl 0263.02033 · doi:10.2307/1996350
[34] 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
[35] Méray, H. C. R., “Remarques sur la nature des quantités définies par la condition de servir de limites à des variables données,” Revue des sociétiés savantes des départments, Section sciences mathématiques, physiques et naturelles (4), vol. 10 (1869), pp. 280-89.
[36] Rust, H., “Operational semantics for timed systems,” Lecture Notes in Computer Science , vol. 3456 (2005), pp. 23-29. · Zbl 1070.68087
[37] Schmieden, C., and D. Laugwitz, “Eine Erweiterung der Infinitesimalrechnung,” Mathematische Zeitschrift , vol. 69 (1958), pp. 1-39. · Zbl 0082.04203 · doi:10.1007/BF01187391
[38] Shenitzer, A., “A topics course in mathematics,” The Mathematical Intelligencer , vol. 9 (1987), pp. 44-52. · Zbl 0623.01029 · doi:10.1007/BF03023955
[39] 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. · Zbl 0010.04902
[40] Street, R., “Update on the efficient reals,” preprint, 2003.
[41] Weber, M. “Leopold Kronecker,” Mathematische Annalen , vol. 43 (1893), pp. 1-25. · doi:10.1007/BF01446613
[42] Weil, A., “Book Review: The mathematical career of Pierre de Fermat,” Bulletin of the American Mathematical Society , vol. 79 (1973), pp. 1138-49. · doi:10.1090/S0002-9904-1973-13354-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.