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The eightfold path to nonstandard analysis

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Published online by Cambridge University Press:  30 March 2017

Mauro Di Nasso
Affiliation:
University of Pisa
Nigel J. Cutland
Affiliation:
University of York
Mauro Di Nasso
Affiliation:
Università degli Studi, Pisa
David A. Ross
Affiliation:
University of Hawaii, Manoa
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Summary

Abstract. This paper consists of a quick introduction to the “hyper-methods” of nonstandard analysis, and of a review of eight different approaches to the subject, which have been recently elaborated by the authors.

Those who follow the noble Eightfold Path are freed from the suffering and are led ultimately to Enlightenment. (Gautama Buddha)

Introduction. Since the originalworks [39, 40] by AbrahamRobinson,many different presentations to the methods of nonstandard analysis have been proposed over the last forty years. The task of combining in a satisfactory manner rigorous theoretical foundations with an easily accessible exposition soon revealed very difficult to be accomplished. The first pioneering work in this direction was W.A.J. Luxemburg's lecture notes [36]. Based on a direct use of the ultrapower construction, those notes were very popular in the “nonstandard” community in the sixties. Also Robinson himself gave a contribution to the sake of simplification, by reformulating his initial typetheoretic approach in a more familiar set-theoretic framework. Precisely, in his joint work with E. Zakon [42], he introduced the superstructure approach, by now the most used foundational framework.

To the authors’ knowledge, the first relevant contribution aimed to make the “hyper-methods” available even at a freshman level, is Keisler's book [33], which is a college textbook for a first course of elementary calculus. There, the principles of nonstandard analysis are presented axiomatically in a nice and elementary form (see the accompanying book [32] for the foundational aspects). Among the more recent works, there are the “gentle” introduction by W.C. Henson [26], R. Goldblatt's lectures on the hyperreals [25], and K.D. Stroyan's textbook [44].

Recently the authors investigated several different frameworks in algebra, topology, and set theory, that turn out to incorporate explicitly or implicitly the “hyper-methods”. These approaches show that nonstandard extensions naturally arise in several quite different contexts of mathematics. An interesting phenomenon is that some of those approaches lead in a straightforward manner to ultrafilter properties that are independent of the axioms of Zermelo-Fraenkel set theory ZFC.

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Publisher: Cambridge University Press
Print publication year: 2006

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References

L.O., Arkeryd, N.J., Cutland, and C.W., Henson (editors), Nonstandard analysis – theory and applications, NATO ASI Series C, vol. 493, Kluwer, A.P., Dordrecht, 1997.
A., Bartoszynski and S., Shelah, There may be noHausdorff ultrafilters, manuscript, 2003.
V., Benci, A construction of a nonstandard universe, Advances of dynamical systems and quantum physics (S., Albeverio et al., editors), World Scientific, Singapore, 1995, pp. 11–21.
V., Benci and M., Di Nasso, Alpha-theory: an elementary axiomatics for nonstandard analysis, Expositiones Mathematicae, vol. 21, (2003), pp. 355–386.Google Scholar
V., Benci and M., Di Nasso, Numerosities of labelled sets: a new way of counting, Advances in Mathematics, vol. 173, (2003), pp. 50–67.Google Scholar
V., Benci and M., Di Nasso, A ring homomorphism is enough to get nonstandard analysis, Bulletin of the Belgian Mathematical Society, vol. 10, (2003), pp. 1–10.Google Scholar
V., Benci and M., Di Nasso, A purely algebraic characterization of the hyperreal numbers, Proceedings of the American Mathematical Society, vol. 133, (2005), pp. 2501–2505.Google Scholar
V., Benci and M., Di Nasso, How to measure the infinite — numerosities and nonstandard analysis, book in preparation.
V., Benci, M., Di Nasso, and M., Forti, Hausdorff nonstandard extensions, Boletim da Sociedade Paranaense de Matemática (3), vol. 20, (2002), pp. 9–20.Google Scholar
A., Blass, A model-theoretic view of some special ultrafilters, Logic colloquium ‘77 (A., Mac-Intyre, L., Pacholski, and J., Paris, editors), North Holland, Amsterdam, 1978, pp. 79–90.Google Scholar
D., Booth, Ultrafilters on a countable set, Annals ofMathematical Logic, vol. 2, (1970/71), pp. 1–24.Google Scholar
C.C., Chang and H.J., Keisler, Model theory, 3rd ed., North-Holland, Amsterdam, 1990.
M., Daguenet-Teissier, Ultrafiltres à la façon de Ramsey, Transactions of the American Mathematical Society, vol. 250, (1979), pp. 91–120.Google Scholar
M., Davis, Applied nonstandard analysis, John Wiley & sons, New York, 1977.
M., Di Nasso, On the foundations of nonstandard mathematics, Mathematica Japonica, vol. 50, (1999), no. 1, pp. 131–160.Google Scholar
M., Di Nasso, ZFC[Ω]: a nonstandard set theory where all sets have nonstandard extensions (abstract), The Bulletin of Symbolic Logic, vol. 7, (2001), p. 138.Google Scholar
M., Di Nasso, An axiomatic presentation of the nonstandardmethods in mathematics, The Journal of Symbolic Logic, vol. 67, (2002), pp. 315–325.Google Scholar
M., Di Nasso and M., Forti, Topological and nonstandard extensions, Monatshefte für Mathematik, vol. 144, (2005), pp. 84–112.Google Scholar
M., Di Nasso and M., Forti, Ultrafilter semirings and nonstandard submodels of the Stone-Čech compactification of the natural numbers, Logic and its applications (A., Blass and Y., Zhang, editors), Contemporary Mathematics, American Mathematical Society, Providence, R.I., 2005.
M., Di Nasso and M., Forti, Hausdorff ultrafilters, Proceedings of the American Mathematical Society, accepted.
F., Diener and M., Diener (editors), Nonstandard analysis in practice, Springer, New York, 1995.
F., Diener and G., Reeb, Analyse non standard, Hermann, Paris, 1989.
R., Engelking, General topology, Polish S.P., Warszawa, 1977.
M., Forti, A functional characterization of complete elementary extensions, submitted. 44 VIERI BENCI, MARCO FORTI, AND MAURO DI NASSO
R., Goldblatt, Lectures on the hyperreals – an introduction to nonstandard analysis, Graduate Texts in Mathematics, vol. 188, Springer, New York, 1998.
C.W., Henson, A gentle introduction to nonstandard extensions, [1], 1997, pp. 1–49.
N., Hindman and D., Strauss, Algebra in the Stone-Čech compactification, W. deGruyter, Berlin, 1998.
K., Hrbacek, Axiomatic foundations for nonstandard analysis, Fundamenta Mathematicae, vol. 98, (1978), pp. 1–19.Google Scholar
K., Hrbacek, Realism, nonstandard set theory, and large cardinals, Annals of Pure and Applied Logic, vol. 109, (2001), pp. 15–48.Google Scholar
V., Kanovei and M., Reeken, Nonstandard analysis, axiomatically, Springer, 2004.
H.J., Keisler, Limit ultrapowers, Transactions of the American Mathematical Society, vol. 107, (1963), pp. 382–408.Google Scholar
H.J., Keisler, Foundations of infinitesimal calculus, Prindle, Weber & Schmidt, Boston, 1976.
H.J., Keisler, Elementary calculus – an infinitesimal approach, 2nd ed., Prindle, Weber & Schmidt, Boston, 1986, [This book is now freely downloadable from the author's homepage: http://www.math.wisc.edu/keisler].
I., Lakatos, Cauchy and the continuum: the significance of non-standard analysis for the history and philosophy of mathematics, Mathematics, science and epistemology (J.W., Worrall and G., Curries, editors), Philosophical Papers, vol. 2, Cambridge University Press, Cambridge, 1978, pp. 43–60.
T., Lindstrom, An invitation to nonstandard analysis, Nonstandard analysis and its applications (N.J., Cutland, editor), London Mathematical Society Student Texts, vol. 10, Cambridge University Press, Cambridge, 1988, pp. 1–105.
W.A.J., Luxemburg, Non-standard analysis, Lecture Notes, CalTech, Pasadena, 1962.
E., Nelson, Internal set theory: a new approach to nonstandard analysis, Bulletin of the American Mathematical Society, vol. 83, (1977), pp. 1165–1198.Google Scholar
A., Robert, Nonstandard analysis, Wiley, 1988.
A., Robinson, Non-standard analysis, Proceedings of the Royal Academy of Amsterdam, vol. 64, (1961), pp. 432–440, (= Indagationes Mathematicae, 23).Google Scholar
A., Robinson, Non-standard analysis, North-Holland, Amsterdam, 1966.
A., Robinson, Compactification of groups and rings and nonstandard analysis, The Journal of Symbolic Logic, vol. 34, (1969), pp. 576–588.Google Scholar
A., Robinson and E., Zakon, A set-theoretical characterization of enlargements, Applications of model theory to algebra, analysis and probability (W.A.J., Luxemburg, editor), Holt, Rinehart & Winston, New York, 1969, pp. 109–122.
S., Shelah, Proper and improper forcing, 2nd ed., Springer, Berlin, 1998.
K.D., Stroyan, Mathematical background – foundations of infinitesimal calculus, Academic Press, New York, 1997, [Freely downloadable from the author's homepage: http://www.math.uiowa.edu/stroyan.].
K.D., Stroyan and W.A.J., Luxemburg, Introduction to the theory of infinitesimals, Academic Press, New York, 1976.

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