Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T06:08:57.524Z Has data issue: false hasContentIssue false
  • English
  • Français

First order theories for nonmonotone inductive definitions: recursively inaccessible and Mahlo

Published online by Cambridge University Press:  12 March 2014

Gerhard Jäger
Gerhard Jäger*
Affiliation:
Nstitut für Informatik und Angewandte Mathematik, Universität Bern, Neubrückstrasse 10 CH-3012 Bern, Switzerland, E-mail: jaeger@iam.unibe.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms à la Richter with recursively inaccessible and Mahlo closure ordinals is given.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1073 - 1089
Copyright
Copyright © Association for Symbolic Logic 2001

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]Aczel, P., An introduction to inductive definitions, Handbook of mathematical logic (Barwise, J., editor), North-Holland, 1977.Google Scholar
[2]Aczel, P. and Richter, W., Inductive definitions and reflecting properties of admissible ordinals, Generalized recursion theory (Fenstad, J. E. and Hinman, P. G., editors), North-Holland, 1974.Google Scholar
[3]Arai, T., Ordinal diagrams for recursively Mahlo universes, Submitted.Google Scholar
[4]Arai, T., Proof theory for theories of ordinals I: recursively Mahlo ordinals, Submitted.Google Scholar
[5]Barwise, J., Admissible sets and structures, Springer, 1975.CrossRefGoogle Scholar
[6]Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., Iterated inductive definitions and subsystems of analysis: Recent proof-theoretical studies, Lecture Notes in Mathematics, no. 897, Springer, 1981.Google Scholar
[7]Feferman, S., A language and axioms for explicit mathematics, Algebra and logic (Crossley, J. N., editor), Lecture Notes in Mathematics, no. 450, Springer, 1975.Google Scholar
[8]Feferman, S., Constructive theories of functions and classes, Logic colloquium '78 (Boffa, M., van Dalen, D., and McAloon, K., editors), North-Holland, 1979.Google Scholar
[9]Jäger, G., Beweistheorie von KPN, Archiv für Mathematische Logik und Grundlagenforschung, vol. 20 (1980).CrossRefGoogle Scholar
[10]Jäger, G., Zur Beweistheorie der Kripke-Platek-Mengenlehre über den natürlichen Zahlen, Archiv für Mathematische Logik und Grundlagenforschung, vol. 22 (1982).Google Scholar
[11]Jäger, G., A well-ordering proof for Feferman's theory T0, Archiv für Mathematische Logik und Grundlagenforschung, vol. 23 (1983).CrossRefGoogle Scholar
[12]Jäger, G., The strength of admissibility without foundation, this Journal, vol. 49 (1984).Google Scholar
[13]Jäger, G., Theories for admissible sets: A unifying approach to proof theory, Bibliopolis, 1986.Google Scholar
[14]Jäger, G., Some proof-theoretic contributions to theories of sets, Logic colloquium '85 (The Paris Logic Group, editor), North-Holland, 1987.Google Scholar
[15]Jäger, G. and Pohlers, W., Eine beweistheoretische Untersuchung von (Δ21-CA)+(BI) und verwandter Systeme, Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, (1982).Google Scholar
[16]Jäger, G. and Studer, T., Extending the system T0 of explicit mathematics: the limit and Mahlo axioms, Annals of Pure and Applied Logic, (to appear).Google Scholar
[17]Moschovakts, Y. N., Elementary induction on abstract structures, North-Holland, 1974.Google Scholar
[18]Pohlers, W., Proof theory and ordinal analysis, Archive for Mathematical Logic, vol. 30 (1991).CrossRefGoogle Scholar
[19]Pohlers, W., A short course in ordinal analysis, Proof theory (Aczel, P., Simmons, H., and Wainer, S., editors), Cambridge University Press, 1992.Google Scholar
[20]Pohlers, W., Pure proof theory: Aims, methods and results, The Bulletin of Symbolic Logic, vol. 2 (1996).CrossRefGoogle Scholar
[21]Rathjen, M., Proof-theoretic analysis of KPM, Archive for Mathematical Logic, vol. 30 (1991).CrossRefGoogle Scholar
[22]Rathjen, M., Fragments of Kripke-Platek set theory with infinity, Proof theory (Aczel, P., Simmons, H., and Wainer, S., editors), Cambridge University Press, 1992.Google Scholar
[23]Rathjen, M., Admissible proof theory and beyond, Logic, methodology and philosophy of science ix (Prawitz, D., Skyrms, B., and Westerståhl, D., editors), North-Holland, 1994.Google Scholar
[24]Rathjen, M., Collapsing functions based on recursively large ordinals: A well-ordering prooffor KPM, Archive for Mathematical Logic, vol. 33 (1994).CrossRefGoogle Scholar
[25]Rathjen, M., Proof theory of reflection, Annals of Pure and Applied Logic, vol. 68 (1994).CrossRefGoogle Scholar
[26]Richter, W., Recursively Mahlo ordinals and inductive definitions, Logic colloquium '69 (Gandy, R. O. and Yates, C. E. M., editors), North-Holland, 1971.Google Scholar
[27]Studer, T., Explicit mathematics: W-type, models, Diploma thesis, Institut für Informatik und angewandte Mathematik, Universität Bern, 1997.Google Scholar