## Notre Dame Journal of Formal Logic

### An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals

#### Abstract

A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).

#### Article information

Source
Notre Dame J. Formal Logic, Volume 53, Number 4 (2012), 557-570.

Dates
First available in Project Euclid: 8 November 2012

https://projecteuclid.org/euclid.ndjfl/1352383232

Digital Object Identifier
doi:10.1215/00294527-1722755

Mathematical Reviews number (MathSciNet)
MR2995420

Zentralblatt MATH identifier
1266.03074

Subjects
Secondary: 03C20: Ultraproducts and related constructions

#### Citation

Borovik, Alexandre; Jin, Renling; Katz, Mikhail G. An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals. Notre Dame J. Formal Logic 53 (2012), no. 4, 557--570. doi:10.1215/00294527-1722755. https://projecteuclid.org/euclid.ndjfl/1352383232

#### References

• [1] A’Campo, N., “A natural construction for the real numbers,” preprint, arXiv:math/0301015v1 [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.
• [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.
• [5] Arkeryd, L., “Nonstandard analysis,” American Mathematical Monthly, vol. 112 (2005), pp. 926–28.
• [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.
• [7] Arthan, R., “The Eudoxus real numbers,” preprint, arXiv:math/0405454v1 [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, http://dx.doi.org/10.1007/s10699-012-9285-8
• [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, http://dx.doi.org/10.1007/s10699-011-9235-X
• [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.
• [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.
• [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.
• [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.
• [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, arXiv:1109.3553v1 [math.LO]
• [17] Goldblatt, R., Lectures on the Hyperreals. An Introduction to Nonstandard Analysis, vol. 188 of Graduate Texts in Mathematics, Springer, New York, 1998.
• [18] Grundhöfer, T., “Describing the real numbers in terms of integers,” Archiv der Mathematik (Basel), vol. 85 (2005), pp. 79–81.
• [19] Hewitt, E., “Rings of real-valued continuous functions, I,” Transactions of the American Mathematical Society, vol. 64 (1948), pp. 45–99.
• [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.
• [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.
• [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.
• [27] Katz, K., and M. Katz, “Stevin numbers and reality,” Foundations of Science, vol. 17 (2012), pp. 109–23.
• [28] Katz, M., and E. Leichtnam, “Commuting and non-commuting infinitesimals,” to appear in American Mathematical Monthly.
• [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, http://dx.doi.org/10.1007/s10670-012-9370-y
• [30] Katz, M., and D. Sherry, “Leibniz’s laws of continuity and homogeneity,” Notices of the American Mathematical Society, vol. 59 (2012), no. 11.
• [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.
• [33] Kunen, K., “Ultrafilters and independent sets,” Transactions of the American Mathematical Society, vol. 172 (1972), pp. 299–306.
• [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.
• [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.
• [37] Schmieden, C., and D. Laugwitz, “Eine Erweiterung der Infinitesimalrechnung,” Mathematische Zeitschrift, vol. 69 (1958), pp. 1–39.
• [38] Shenitzer, A., “A topics course in mathematics,” The Mathematical Intelligencer, vol. 9 (1987), pp. 44–52.
• [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.
• [40] Street, R., “Update on the efficient reals,” preprint, 2003.
• [41] Weber, M. “Leopold Kronecker,” Mathematische Annalen, vol. 43 (1893), pp. 1–25.
• [42] Weil, A., “Book Review: The mathematical career of Pierre de Fermat,” Bulletin of the American Mathematical Society, vol. 79 (1973), pp. 1138–49.