Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T18:49:33.940Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Sam Sanders
Affiliation:
University of Ghent, Department of Pure Mathematics and Computer Algebra, Galglaan 2, B-9000 Gent, Belgium, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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