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Nonstandard Arithmetic and Reverse Mathematics

Published online by Cambridge University Press:  15 January 2014

H. Jerome Keisler
H. Jerome Keisler*
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, USA. E-mail: keisler@math.wisc.edu
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Abstract

We show that each of the five basic theories of second order arithmetic that play a central role in reverse mathematics has a natural counterpart in the language of nonstandard arithmetic. In the earlier paper [3] we introduced saturation principles in nonstandard arithmetic which are equivalent in strength to strong choice axioms in second order arithmetic. This paper studies principles which are equivalent in strength to weaker theories in second order arithmetic.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 12 , Issue 1 , March 2006 , pp. 100 - 125
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1] Chang, C. C. and Keisler, H. J., Model theory, third ed., Elsevier, 1990.Google Scholar
[2] Enayat, A., From bounded arithmetic to second order arithmetic via automorphisms, Logic in Tehran (Enayat, A., Kalantari, I., and Moniri, M., editors), Lecture Notes in Logic, vol. 26, ASL and AK Peters, 2006, To appear. Available online at http://academic2.american.edu/~enayat.CrossRefGoogle Scholar
[3] Henson, C.W., Kaufmann, M., and Keisler, H. J., The strength of nonstandard methods in arithmetic, The Journal of Symbolic Logic, vol. 49 (1984), pp. 1039–1058.Google Scholar
[4] Henson, C. W. and Keisler, H. J., On the strength of nonstandard analysis, The Journal of Symbolic Logic, vol. 51 (1986), pp. 377–386.Google Scholar
[5] Keisler, H. J., The strength of nonstandard analysis, Available online at http://www.math.wisc.edu/~keisler, To appear.Google Scholar
[6] MacDowell, R. and Specker, E., Modelle der Arithmetik, Infinitistic methods, Pergamon Press, 1961, pp. 257–263.Google Scholar
[7] Simpson, S. G., Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, 1999.Google Scholar
[8] Tanaka, K., The self-embedding theorem of WKL0 and a non-standard method, Annals of Pure and Applied Logic, vol. 84 (1997), pp. 41–49.CrossRefGoogle Scholar