The strength of countable saturation
- 143 Downloads
In earlier work we introduced two systems for nonstandard analysis, one based on classical and one based on intuitionistic logic; these systems were conservative extensions of first-order Peano and Heyting arithmetic, respectively. In this paper we study how adding the principle of countable saturation to these systems affects their proof-theoretic strength. We will show that adding countable saturation to our intuitionistic system does not increase its proof-theoretic strength, while adding it to the classical system increases the strength from first- to full second-order arithmetic.
KeywordsProof theory Nonstandard analysis Arithmetic in all finite types Saturation principles
Mathematics Subject Classification03F10 03F25 03F50 11U10 26E35
- 1.Avigad, J., Feferman, S.: Gödel’s functional (“Dialectica”) interpretation. In: Buss, S.R. (ed.) Handbook of Proof Theory, Volume 137 of Studies in Logic and the Foundations of Mathematics, pp. 337–405. North-Holland, Amsterdam (1998)Google Scholar
- 2.Awodey, S., Eliasson, J.: Ultrasheaves and double negation. Notre Dame J. Form. Log. 45(4), 235–245 (2004) (electronic)Google Scholar
- 4.Escardó, M., Oliva, P.: The Herbrand functional interpretation of the double negation shift. J. Symb. Log. arXiv:1410.4353 (2015)
- 5.Geyer, C.J.: Radically elementary probability and statistics. Technical Report No. 657, School of Statistics, University of Minnesota. http://www.stat.umn.edu/geyer/nsa/o.pdf (2007)
- 9.Nelson, E.: Radically Elementary Probability Theory, Volume 117 of Annals of Mathematics Studies. Princeton University Press, Princeton (1987)Google Scholar
- 11.Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Proceedings of Symposium Pure Mathematics, Vol. 5, pp. 1–27. American Mathematical Society, Providence, RI (1962)Google Scholar
- 12.Troelstra, A.S. (eds.): Metamathematical investigation of intuitionistic arithmetic and analysis. Lecture Notes in Mathematics, Vol. 344. Springer, Berlin (1973)Google Scholar
- 13.Troelstra, A.S., van Dalen, D.: Constructivism in mathematics. Vol. 2, volume 123 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1988)Google Scholar
- 14.van den Berg, B., Briseid, E.: Weak systems for nonstandard arithmetic (2017) (in preparation)Google Scholar
- 16.van den Berg, B., Sanders, S.: Reverse mathematics and parameter-free transfer. arXiv:1409.6881 (2017)
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.