Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-19T06:00:19.114Z Has data issue: false hasContentIssue false
  • English
  • Français

Fragment of Nonstandard Analysis with a Finitary Consistency Proof

Published online by Cambridge University Press:  15 January 2014

Michal Rössler  and
Emil Jeřábek
Michal Rössler
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Prague, 115 67, Czech Republic, E-mail: rossler@math.cas.czE-mail: jerabek@math.cas.cz
Emil Jeřábek
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Prague, 115 67, Czech Republic, E-mail: rossler@math.cas.czE-mail: jerabek@math.cas.cz
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a nonstandard arithmetic NQA− based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by NQA+), with a weakened external open minimization schema. A finitary consistency proof for NQA− formalizable in PRA is presented. We also show interesting facts about the strength of the theories NQA−and NQA+; NQA−is mutually interpretable with IΔ0 + EXP, and on the other hand, NQA+interprets the theories IΣ1 and WKL0.

Type
Communications
Information
Bulletin of Symbolic Logic , Volume 13 , Issue 1 , March 2007 , pp. 54 - 70
Copyright
Copyright © Association for Symbolic Logic 2007

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] Ballard, David and Hrbáček, Karel, Standard foundations for nonstandard analysis, The Journal of Symbolic Logic, vol. 57 (1992), no. 2, pp. 741–748.Google Scholar
[2] Chuaqui, Rolando and Suppes, Patrick, Free-variable axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof, The Journal of Symbolic Logic, vol. 60 (1995), no. 1, pp. 122–159.Google Scholar
[3] Feferman, Solomon, Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 35–92.Google Scholar
[4] Fernandes, António M. and Ferreira, Fernando, Groundwork for weak analysis, The Journal of Symbolic Logic, vol. 67 (2002), no. 2, pp. 557–578.Google Scholar
[5] Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic, Springer-Verlag, 1993.Google Scholar
[6] Mycielski, Jan, Locally finite theories, The Journal of Symbolic Logic, vol. 51 (1986), pp. 59–62.Google Scholar
[7] Nelson, Edward, Internal set theory: A new approach to nonstandard analysis, Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 1165–1198.Google Scholar
[8] Pudlák, Pavel, Cuts, consistency statements and interpretability, The Journal of Symbolic Logic, vol. 50 (1985), pp. 423–441.CrossRefGoogle Scholar
[9] Robinson, Abraham, Non-standard analysis, North-Holland Publishing Company, 1966.Google Scholar
[10] Shoenfield, Joseph R., Mathematical logic, Addison-Wesley Publishing Company, 1967.Google Scholar
[11] Sochor, Antonín, Klasická matematická logika, Karolinum, Prague, 2001, in Czech.Google Scholar
[12] Suppes, Patrick and Chuaqui, Rolando, A finitarily consistent free-variable positive fragment of infinitesimal analysis, Notas de Logica Matematica, vol. 38 (1993), pp. 1–59, Proceedings of the IX Latin American Symposium on Mathematical Logic.Google Scholar
[13] Tait, William W., Finitism, Journal of Philosophy, vol. 78 (1981), pp. 524–564.Google Scholar
[14] Visser, Albert, An overview of interpretability logic, Advances in modal logic '96 (Kracht, M., de Rijke, M., Wansing, H., and Zakharyaschev, M., editors), CSLI, Stanford, 1998, pp. 307–359.Google Scholar
[15] Vopěnka, Petr, Mathematics in the Alternative Set Theory, Teubner, Leipzig, 1979.Google Scholar
[16] Zach, Richard, The practice of finitism: Epsilon calculus and consistency proofs in Hilbert's program, Syntese, vol. 137 (2003), pp. 211–259.Google Scholar