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An axiomatic presentation of the nonstandard methods in mathematics

Published online by Cambridge University Press:  12 March 2014

Mauro Di Nasso
Mauro Di Nasso*
Affiliation:
Università di Pisa, Dipartimento di Matematica Applicata Via Bonanno 25/B, 55126 Pisa, Italy, E-mail: [email protected]
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Abstract

A nonstandard set theory *ZFC is proposed that axiomatizes the nonstandard embedding *. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. *ZFC is a conservative extension of ZFC.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 315 - 325
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1]Ballard, D. and Hrbàček, K., Standard foundations for nonstandard analysis, this Journal, vol. 57 (1992), pp. 741–748.Google Scholar
[2]Boffa, M., Forcing el négation de l'axiome de fondement, Mémoires de l'Académie des Sciences de Belgique, vol. XL (1972), no. 7.Google Scholar
[3]Chang, C. C. and Keisler, H. J., Model Theory, North-Holland, 1990, 3rd edition.Google Scholar
[4]Nasso, M. Di, Elementary embeddings of the universe in a nonwellfounded context, in preparation.Google Scholar
[5]Nasso, M. Di, *ZFC: An axiomatic * approach to nonstandard methods, Comptes Rendus de l'Académie des Sciences, Série I, vol. 324 (1997), no. 9, pp. 963–967.Google Scholar
[6]Nasso, M. Di, Pseudo-superstructures as nonstandard universes, this Journal, vol. 63 (1998), pp. 222–236.Google Scholar
[7]Nasso, M. Di, On the foundations of nonstandard mathematics, Mathematica Japonica, vol. 50 (1999), pp. 131–160.Google Scholar
[8]Hodges, W., Model Theory, Cambridge University Press, 1993.CrossRefGoogle Scholar
[9]Hrbàček, K., Axiomatic foundations for nonstandard analysis, Fundamenta Mathematicae, vol. 98 (1978), pp. 1–19.CrossRefGoogle Scholar
[10]Hurd, A. E. and Loeb, P. A., An introduction to nonstandard analysis, Academic Press, 1985.Google Scholar
[11]Jech, T., Set Theory, Academic Press, 1978.Google Scholar
[12]Kawai, T., Nonstandard analysis by axiomatic method, Proceedings of the Southeast Asian Conference on Logic (Chang, C. T. and Wicks, M. J., editors), North-Holland, 1981, pp. 55–76.Google Scholar
[13]Keisler, H. J. and Morley, M., Elementary extensions of models of set theory, Israel Journal of Mathematics, vol. 6 (1968), pp. 49–65.CrossRefGoogle Scholar
[14]Kunen, K., Set theory: An introduction to independence proofs, North-Holland, 1980.Google Scholar
[15]Robinson, A. and Zakon, E., A set-theoretical characterization of enlargements, Applications of model theory to algebra, analysis and probability, Holt, Rinehart and Winston, 1969, pp. 109–122.Google Scholar