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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: [email protected]
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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

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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