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Saturation and Σ2-transfer for ERNA

Published online by Cambridge University Press:  12 March 2014

Chris Impens and
Sam Sanders
Chris Impens
Affiliation:
University of Ghent, Department of Pure Mathematics and Computer Algebra, Galglaan 2, B-9000 Gent, Belgium, E-mail: ci@cage.ugent.be
Sam Sanders
Affiliation:
University of Ghent, Department of Pure Mathematics and Computer Algebra, Galglaan 2, B-9000 Gent, Belgium, E-mail: sasander@cage.ugent.be
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Abstract

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. It is built on a previous system by Rolando Chuaqui and Patrick Suppes, which was recently reconsidered by Michal Rössler and Emil Jeřábek. A Π1-transfer principle has already been added to ERNA and the consistency of the resulting theory proved in PRA. Here, we equip ERNA with Σ2-transfer and a saturation principle, while keeping the consistency proof inside PRA. We show that the extended theory allows for generalized transfer, a basic tool of nonstandard analysis, and interprets several strong theories, like BΣ2 and IΣ2.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009, pp. 901 - 913
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Buss, Samuel R., An introduction to proof theory, Handbook of proof theory, Studies in Logic and the Foundations of Mathematics, vol. 137, North-Holland, Amsterdam, 1998, pp. 1–78.CrossRefGoogle Scholar
[2]Chang, Chen Chung and Keisler, H. Jerome, Model theory, 3 ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, 1990.Google Scholar
[3]Chuaqui, Rolando and Suppes, Patrick, Free-variable Axiomatic Foundations of Infinitesimal Analysis: A Fragment with Finitary Consistency Proof, this Journal, vol. 60 (1995), pp. 122–159.Google Scholar
[4]Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
[5]Impens, Chris and a, Transfer and a supremum principle for ERNA, this Journal, vol. 73 (2008), no. 2, pp. 689–710.Google Scholar
[6]Kanovei, Vladimir and Reeken, Michael, Nonstandard analysis, axiomatically, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.CrossRefGoogle Scholar
[7]Rössler, Michal and Jeřábek, Emil, Fragment of nonstandard analysis with a finitary consistency proof, The Bulletin of Symbolic Logic, vol. 13 (2007), pp. 54–70.CrossRefGoogle Scholar
[8]Shoenfield, Joseph R., Mathematical logic, Addison-Wesley, Reading, Mass., 1967.Google Scholar
[9]Sommer, Richard and Suppes, Patrick, Finite models of elementary recursive nonstandard analysis, Notas de la Sociedad Matematica de Chile, vol. 15 (1996), pp. 73–95.Google Scholar
[10]Sommer, Richard and Suppes, Patrick, Dispensing with the continuum, Journal of Mathematical Psychology, vol. 41 (1997), no. 1, pp. 3–10.CrossRefGoogle Scholar
[11]Stroyan, Keith D. and Luxemburg, Willem A. J., Introduction to the theory of infinitesimals, Pure and Applied Mathematics, no. 72, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976.Google Scholar
[12]Suppes, Patrick and Chuaqui, Rolando, A finitarily consistent free-variable positive fragment of infinitesimal analysis, Proceedings of the IXth Latin American Symposium on Mathematical Logic Notas de Logica Matematica, vol. 38 (1993), pp. 1–59.Google Scholar