Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T20:58:31.106Z Has data issue: false hasContentIssue false
  • English
  • Français

LIFTING PROOF THEORY TO THE COUNTABLE ORDINALS: ZERMELO-FRAENKEL SET THEORY

Published online by Cambridge University Press:  25 June 2014

TOSHIYASU ARAI
TOSHIYASU ARAI*
Affiliation:
GRADUATE SCHOOL OF SCIENCE CHIBA UNIVERSITY CHIBA, 263-8522, JAPANE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound on definable countable ordinals in Zermelo-Fraenkel set theory ZF.

Keywords

Provably countableSkolem hullcollapsing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 325 - 354
Copyright
Copyright © The Association for Symbolic Logic 2014 

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

Arai, T., Intuitionistic fixed point theories over set theories, submitted, arXiv: 1312.1133.Google Scholar
Arai, T., Proof theory for theories of ordinals III: ${\rm{\Pi }}_N $ -reflection, Gentzen’s Centenary: The Quest of Consistency, Springer, to appear.Google Scholar
Arai, T., Proof theory for theories of ordinals I: recursively Mahlo ordinals, Annals of Pure and Applied Logic, vol. 122 (2003), pp. 1–85.Google Scholar
Arai, T., Proof theory for theories of ordinals II: ${\rm{\Pi }}_3 $-reflection, Annals of Pure and Applied Logic, vol. 129 (2004), pp. 39–92.Google Scholar
Arai, T., A sneak preview of proof theory of ordinals, Annals of Japan Association for Philosophy of Sciences, vol. 20 (2012), pp. 29–47.Google Scholar
Arai, T., Proof theory of weak compactness, Journal of Mathematical Logic, vol. 13 (2013), p. 26.Google Scholar
Barwise, J., Admissible sets and structures, Springer, Berlin, Heidelberg, 1975.Google Scholar
Buchholz, W., A simplified version of local predicativity, Proof theory (Simmons, H., Aczel, P. H. G., and Wainer, S. S., editors), Cambridge University Press, Cambridge, 1992, pp. 115–147.Google Scholar
Buchholz, W., An intuitionistic fixed point theory, Archive for Mathematical Logic, vol. 37 (1997), pp. 21–27.Google Scholar
Buchholz, W., Finitary treatment of operator controlled derivations, Mathematical Logic Quarterly, vol. 47 (2001), pp. 363–396.Google Scholar
Buchholz, W., Relating ordinals to proofs in a perspicuous way, Reflections on the foundations of mathematics (Sommer, R., Sieg, W. and Talcott, C., editors), Association of Symbolic Logic, 2002, pp. 37–59.Google Scholar
Devlin, K. J., Constructibility, Springer, Berlin, Heidelberg, 1984.Google Scholar
Pohlers, W., Proof theory the first step into impredicativity, Springer, Berlin, Heidelberg, 2009.Google Scholar
Rathjen, M., Proof-theoretic analysis of KPM, Archive for Mathematical Logic, vol. 30 (1991), pp. 377–403.Google Scholar
Rathjen, M., Proof theory of refleection, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 181–224.Google Scholar
Rathjen, M., An ordinal analysis of parameter free $\pi _2^1 $-comprehension, Archive for Mathematical Logic, vol. 44 (2005), pp. 263–362.Google Scholar
Rathjen, M., An ordinal analysis of stability, Archive for Mathematical Logic, vol. 44 (2005), pp. 1–62.Google Scholar
Richter, W. H. and Aczel, P., Inductive definitions and reflecting properties of admissible ordinals, Generalized recursion theory (Fenstad, J. E. and Hinman, P. G., editors), Studies in Logic, vol. 79, North-Holland, Amsterdam, 1974, pp. 301–381.Google Scholar
Schütte, K., Proof theory, Springer, Berlin, Heidelberg, 1977.CrossRefGoogle Scholar