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Developments in Constructive Nonstandard Analysis

Published online by Cambridge University Press:  15 January 2014

Erik Palmgren
Erik Palmgren*
Affiliation:
Department of Mathematics, Uppsala University, P.O. BOX 480, S-751 06 Uppsala, Sweden.E-mail: palmgren@math.uu.se
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Abstract

We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine–Borel theorem, the Cauchy–Peano existence theorem for ordinary differential equations and the exact intermediate-value theorem, while it at the same time provides constructive results for concrete statements. A nonstandard measure theory which is considerably simpler than that of Bishop and Cheng is developed within this context.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 4 , Issue 3 , September 1998 , pp. 233 - 272
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1] Bishop, E., Foundations of constructive analysis, McGraw-Hill, 1967.Google Scholar
[2] Bishop, E. and Bridges, D. S., Constructive analysis, Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar
[3] Blass, A., Two closed categories of filters, Fundamenta Mathematicae, vol. 94 (1977), pp. 129–143.Google Scholar
[4] Bridges, D. S. and Richman, F., Varieties of constructive mathematics, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
[5] Cartier, P. and Perrin, Y., Integration over finite sets, in [9].Google Scholar
[6] Chuaqui, R. and Suppes, P., Free-variable axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof, this Journal, vol. 60 (1995), pp. 122–159.Google Scholar
[7] Coquand, T., A formal space of ultra filters, Annals of Pure and Applied Logic, to appear.Google Scholar
[8] Delfini, P. and Lobry, C., The vibrating string, in [9].Google Scholar
[9] Diener, F. and Diener, M. (editors), Nonstandard analysis in practice, Springer-Verlag, Berlin, 1995.Google Scholar
[10] Diener, F. and Reeb, G., Analyse non standard, Hermann, Paris, 1989.Google Scholar
[11] Harthong, J., Éléments pour une théorie du continu, Astérisque, vol. 109-110 (1983), pp. 235–244.Google Scholar
[12] Kanovei, V.G., The correctness of Euler's method for the factorization of the sine function into an infinite product, Russian Mathematical Surveys, vol. 43 (1988), pp. 65–94.Google Scholar
[13] Luxemburg, W. A. J., What is nonstandard analysis?, American Mathematical Monthly, vol. 80 (1973), pp. 38–67.Google Scholar
[14] Lane, S. Mac and Moerdijk, I., Sheaves in geometry and logic, Springer-Verlag, Berlin, 1992.Google Scholar
[15] Martin-Löf, P., Intuitionistic type theory, Bibliopolis, Naples, 1984.Google Scholar
[16] Martin-Löf, P., Mathematics of infinity, COLOG-88 Computer logic (Martin-Löf, P. and Mints, G. E., editors), Lecture Notes in Computer Science, vol. 417, Springer-Verlag, Berlin, 1990, pp. 146–197.CrossRefGoogle Scholar
[17] Moerdijk, I., A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37–51.CrossRefGoogle Scholar
[18] Moerdijk, I. and Palmgren, E., Minimal models of Heyting arithmetic, this Journal, vol. 62 (1997), pp. 1448–1460.Google Scholar
[19] Moerdijk, I. and Reyes, G. E., Smooth infinitesimal analysis, Springer-Verlag, 1991.CrossRefGoogle Scholar
[20] Mycielski, J., Analysis without actual infinity, this Journal, vol. 46 (1981), pp. 625–633.Google Scholar
[21] Nelson, E., Radically elementary probability theory, Princeton University Press, 1987.Google Scholar
[22] Palmgren, E., A constructive approach to nonstandard analysis, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 297–325.Google Scholar
[23] Palmgren, E., Constructive nonstandard analysis, Méthodes et analyse non standard (Pétry, A., editor), Cahiers du Centre de Logique, vol. 9, Academia, Louvain-la-Neuve, 1996, pp. 369–397.Google Scholar
[24] Palmgren, E., Sheaf-theoretic foundation for nonstandard analysis, Annals of Pure and Applied Logic, vol. 85 (1997), pp. 69–86.CrossRefGoogle Scholar
[25] Richman, F., Polynomials and linear transformations, Linear Algebra and its Applications, vol. 131 (1990), pp. 131–137.Google Scholar
[26] Robinson, A., Non-standard analysis, Princeton University Press, 1996, reprint of the 1973 edition with a foreword by W. A. J. Luxemburg.Google Scholar
[27] Schmieden, C. and Laugwitz, D., Eine Erweiterung der Infinitesimalrechnung, Mathematisches Zeitschrift, vol. 69 (1958), pp. 1–39.Google Scholar
[28] Stroyan, K. D. and Luxemburg, W. A. J., Introduction to the theory of infinitesimals, Academic Press, New York, 1976.Google Scholar
[29] Wattenberg, F., Nonstandard analysis and constructivism?, Studia Logica, vol. 47 (1988), pp. 303–309.Google Scholar