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ERNA and Friedman's Reverse Mathematics

Published online by Cambridge University Press:  12 March 2014

Sam Sanders
Sam Sanders*
Affiliation:
University of Ghent, Department of Mathematics, Krijgslaan 281, B-9000 Gent, Belgium, E-mail: [email protected]
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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. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA0 from Reverse Mathematics (see [21] and [22]) can be ‘pushed down’ into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality ‘up to infinitesimals’. It turns out that ERNA plays the role of RCA0 and that transfer for universal formulas corresponds to WKL.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 637 - 664
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Avigad, Jeremy, Weak theories of nonstandard arithmetic and analysis, Reverse mathematics 2001, Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, La Jolla, CA, 2005, pp. 19–46.Google Scholar
[2]Bishop, Errett and Bridges, Douglas S., Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar
[3]Bridges, Douglas S., Constructive functional analysis, Research Notes in Mathematics, vol. 28, Pitman Publishing, London, San Francisco and Melbourne, 1979.Google Scholar
[4]Douglass, S. Bridges and Vîtă, Luminita Simona, Techniques of constructive analysis, Universitext, Springer, 2006.Google Scholar
[5]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
[6]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
[7]Davidson, Kenneth R. and Donsig, Allan P., Real analysis with real applications, Prentice Hall, Upper Saddle River, NJ, 2002.Google Scholar
[8]Friedman, Harvey, Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), vol. 1, Canadian Mathematics Congress, Montreal, Quebec, 1975, pp. 235–242.Google Scholar
[9]Friedman, Harvey, Systems of second order arithmetic with restricted induction, I & II (abstracts), this Journal, vol. 41 (1976), pp. 557–559.Google Scholar
[10]Hurd, Albert E. and Loeb, Peter A., An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press Inc., Orlando, FL, 1985.Google Scholar
[11]Impens, Chris and Sanders, Sam, Erna at work, The strength of nonstandard analysis, Springer Wien NewYork Vienna, 2007, pp. 64–75.CrossRefGoogle Scholar
[12]Impens, Chris, Transfer and a supremum principle for ERNA, this Journal, vol. 73 (2008), pp. 689–710.Google Scholar
[13]Impens, Chris, Saturation and Σ2-transfer for ERNA, this Journal, vol. 74 (2009), pp. 901–913.Google Scholar
[14]Ishihara, Hajime, Constructive reverse mathematics: compactness properties, From sets and types to topology and analysis, Oxford Logic Guides, vol. 48, Oxford University Press, Oxford, 2005, pp. 245–267.CrossRefGoogle Scholar
[15]Ishihara, Hajime, Reverse mathematics in Bishop's constructive mathematics, Philosophia Scientiae (Cahier Spécial), vol. 6 (2006), pp. 43–59.CrossRefGoogle Scholar
[16]Keisler, H. Jerome, Nonstandard arithmetic and reverse mathematics, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 1, pp. 100–125.CrossRefGoogle Scholar
[17]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
[18]Sakamoto, Nobuyuki and Yokoyama, Keita, The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic. Archive in Mathematical Logic, vol. 46 (2007), no. 5–6, pp. 465–480.CrossRefGoogle Scholar
[19]Sanders, Sam, More infinity for a better finitism, Annals of Pure and Applied Logic, vol. 161 (2010), no. 12, pp. 1525–1540.CrossRefGoogle Scholar
[20]Simpson, Stephen G., Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?, this Journal, vol. 49 (1984), no. 3, pp. 783–802.Google Scholar
[21]Simpson, Stephen G. (Editor), Reverse mathematics 2001, Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, La Jolla, CA, 2005.Google Scholar
[22]Simpson, Stephen G., Subsystems of second order arithmetic, 2 ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
[23]Sommer, Richard and Suppes, Patrick, Finite models of elementary recursive nonstandard analysis. Notas de la Sociedad Mathematica de Chile, vol. 15 (1996), pp. 73–95.Google Scholar
[24]Sommer, Richard, Dispensing with the continuum, Journal of Mathematical Psychology, vol. 41 (1997), pp. 3–10.CrossRefGoogle Scholar
[25]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
[26]Yokoyama, Keita, Complex analysis in subsystems of second order arithmetic, Archive for Mathematical Logic, vol. 46 (2007), no. 1, pp. 15–35.CrossRefGoogle Scholar