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Transfer and a supremum principle 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], URL: http://cage.ugent.be/~ci
Sam Sanders
Affiliation:
University of Ghent, Department of Pure Mathematics and Computer Algebra, Galglaan 2, B-9000 Gent, Belgium, E-mail: [email protected], URL: http://cage.ugent.be/~sasander
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Abstract

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Patrick Suppes and Richard Sommer, who also proved its consistency inside PRA. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes, of which Michal Rössler and Emil Jeřábek have recently proposed a weakened version. We add a Πı-transfer principle to ERNA and prove the consistency of the extended theory inside PRA. In this extension of ERNA a Σı-supremum principle ‘up-to-infinitesimals’, and some well-known calculus results for sequences are deduced. Finally, we prove that transfer is ‘too strong’ for finitism by reconsidering Rössler and Jeřábek's conclusions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008 , pp. 689 - 710
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Buss, Samuel, Proof theory, Elsevier, 1998.Google Scholar
[2]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
[3]Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic. Springer, 1998.Google Scholar
[4]Impens, Chris and Sanders, Sam, ERNA at work, The strength of nonstandard analysis (van den Berg, Imme and Neves, Vitor, editors), Springer, 2007, pp. 64–75.CrossRefGoogle Scholar
[5]Kanovei, Vladimir and Reeken, Michael, Nonstandard analysis, axiomatically, Springer, 2004.CrossRefGoogle Scholar
[6]Kohlenbach, Ulrich, Things that can and things that can't be done in PRA, Annals of Pure and Applied Logic, vol. 102 (2000), pp. 223–245.CrossRefGoogle Scholar
[7]Parsons, Charles, On a number-theoretic choice scheme and its relation to induction, Intuitionism and proof theory (Myhill, J., Kino, A., and Vesley, R. E., editors), North-Holland, 1970, pp. 459–473.Google Scholar
[8]Parsons, Charles, On n-quantifier induction, this Journal, vol. 37 (1972), pp. 466–482.Google Scholar
[9]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
[10]Sommer, Richard and Suppes, Patrick, Finite models of elementary recursive nonstandard analysis, Notas de la Sociedad Matemdtica de Chile, vol. 15 (1996), pp. 73–95.Google Scholar
[11]Stroyan, Keith D. and Luxemburg, Willem A. J., Introduction to the theory of infinitesimals, Academic Press, 1976.Google Scholar
[12]Tait, William W., Finitism, The Journal of Philosophy, vol. 78 (1981), pp. 524–564.CrossRefGoogle Scholar