Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-16T05:20:17.896Z Has data issue: false hasContentIssue false
  • English
  • Français

RELATIVIZING OPERATIONAL SET THEORY

Published online by Cambridge University Press:  10 October 2016

GERHARD JÄGER
GERHARD JÄGER*
Affiliation:
GERHARD JÄGER, INSTITUT FÜR INFORMATIK UNIVERSITÄT BERN, NEUBRÜCKSTRASSE 10 CH-3012 BERN, SWITZERLANDE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a way of relativizing operational set theory that also takes care of application. After presenting the basic approach and proving some essential properties of this new form of relativization we turn to the notion of relativized regularity and to the system OST (LR) that extends OST by a limit axiom claiming that any set is element of a relativized regular set. Finally we show that OST (LR) is proof-theoretically equivalent to the well-known theory KPi for a recursively inaccessible universe.

Keywords

03B3003B4003E2003F03operational set theoryKripke–Platek set theoryproof theory
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 22 , Issue 3 , September 2016 , pp. 332 - 352
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

Barwise, K. J.,Admissible Sets and Structures, Perspectives in Mathematical Logic, vol. 7, Springer, Berlin, Heidelberg, New York, 1975.CrossRefGoogle Scholar
Beeson, M. J., Foundations of Constructive Mathematics: Metamathematical Studies, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3/6, Springer, Berlin, Heidelberg, New York, Tokyo, 1985.CrossRefGoogle Scholar
Cantini, A., Extending constructive operational set theory by impredicative principles . Mathematical Logic Quarterly, vol. 57 (2011), no. 3, pp. 299–322.CrossRefGoogle Scholar
Cantini, A. and Crosilla, L., Constructive set theory with operations , Logic Colloquium 2004 (Andretta, A., Kearnes, K., and Zambella, D., editors), Lecture Notes in Logic, vol. 29, Cambridge University Press, Cambridge, 2007, pp. 47–83.CrossRefGoogle Scholar
Cantini, A. and Crosilla, L., Elementary constructive operational set theory , Ways of Proof Theory (Schindler, R., editor), Ontos Mathematical Logic, vol. 2, De Gruyter, Frankfurt, 2010, pp. 199–240.CrossRefGoogle Scholar
Feferman, S., A language and axioms for explicit mathematics , Algebra and Logic (Crossley, J. N., editor), Lecture Notes in Mathematics, vol. 450, Springer, Berlin, Heidelberg, New York, 1975, pp. 87–139.CrossRefGoogle Scholar
[7] Feferman, S., Notes on operational set theory, I. Generalization of “small” large cardinals in classical and admissible set theory, Technical Notes , 2001.Google Scholar
Feferman, S., Operational set theory and small large cardinals . Information and Computation, vol. 207 (2009), pp. 971–979.CrossRefGoogle Scholar
[9] Jäger, G., Die konstruktible Hierarchie als Hilfsmittel zur beweistheoretischen Untersuchung von Teilsystemen der Mengenlehre und Analysis, Ph.D. thesis, Mathematisches Institut, Universität München, 1979.Google Scholar
Jäger, G., A well-ordering proof for Feferman’s theory T0 . Archiv für Mathematische Logik und Grundlagenforschung, vol. 23 (1983), no. 1, pp. 65–77.CrossRefGoogle Scholar
Jäger, G., Theories for Admissible Sets: A Unifying Approach to Proof Theory , Studies in Proof Theory, Lecture Notes, vol. 2, Bibliopolis, Napoli, 1986.Google Scholar
Jäger, G., On Feferman’s operational set theory OST. Annals of Pure and Applied Logic, vol. 150 (2007), no. 1–3, pp. 19–39.CrossRefGoogle Scholar
Jäger, G., Full operational set theory with unbounded existential quantification and power set . Annals of Pure and Applied Logic, vol. 160 (2009), no. 1, pp. 33–52.CrossRefGoogle Scholar
Jäger, G., Operations, sets and classes , Logic, Methodology and Philosophy of Science—Proceedings of the Thirteenth International Congress (Glymour, C., Wei, W., and Westerståhl, D., editors), College Publications, London, 2009, pp. 74–96.Google Scholar
Jäger, G., Operational closure and stability . Annals of Pure and Applied Logic, vol. 164 (2013), no. 7–8, 813–821.CrossRefGoogle Scholar
Jäger, G. and Pohlers, W., Eine beweistheoretische Untersuchung von $\left( {{\rm{\Delta }}_2^1 - CA} \right) + \left( {BI} \right)$ und verwandter Systeme . Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse (1982), pp. 1–28.Google Scholar
Jäger, G. and Zumbrunnen, R., About the strength of operational regularity , Logic, Construction, Computation (Berger, U., Diener, H., Schuster, P., and Seisenberger, M., editors), Ontos Mathematical Logic, vol. 3, De Gruyter, Frankfurt, 2012, pp. 305–324.CrossRefGoogle Scholar
[18] Jäger, G. and Zumbrunnen, R., Explicit mathematics and operational set theory: some ontological comparisons, this Bulletin, vol. 20 (2014), no. 3, pp. 275–292.Google Scholar
Kunen, K., Set Theory. An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, Elsevier, Amsterdam, New York, 1980.Google Scholar
[20] Probst, D., Pseudo-hierarchies in admissible set theory without foundation and explicit mathematics, Ph.D. thesis, Institut für Informatik und angewandte Mathematik, Universität Bern, 2005.Google Scholar
Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, I, Studies in Logic and the Foundations of Mathematics, vol. 121, Elsevier, Amsterdam, New York, 1988.Google Scholar
[22] Zumbrunnen, R., Contributions to operational set theory, Ph.D. thesis, Institut für Informatik und angewandte Mathematik, Universität Bern, 2013.Google Scholar