Hostname: page-component-6bf8c574d5-8gtf8 Total loading time: 0 Render date: 2025-02-22T10:58:09.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotone inductive definitions in explicit mathematics

Published online by Cambridge University Press:  12 March 2014

Michael Rathjen
Michael Rathjen*
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The context for this paper is Feferman's theory of explicit mathematics, T0. We address a problem that was posed in [6]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T0 + MID, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce that MID, when adjoined to classical T0, leads to a much stronger theory than T0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 1 , March 1996 , pp. 125 - 146
Copyright
Copyright © Association for Symbolic Logic 1996

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, Amsterdam, 1977, pp. 739–782.CrossRefGoogle Scholar
[2]Beeson, M., Foundations of constructive mathematics, Springer Verlag, Berlin, 1985.CrossRefGoogle Scholar
[3]Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., Iterated inductive definitions and subsystems of analysis, Lecture notes in mathematics, no. 897, Springer Verlag, Berlin, 1981, pp. 78–142.Google Scholar
[4]Feferman, S., A language and axioms for explicit mathematics, Algebra and logic (Crossley, J. N., editor), Lecture Notes in Mathematics, no. 450, Springer Verlag, Berlin, 1975, pp. 87–139.CrossRefGoogle Scholar
[5]Feferman, S., Constructive theories of functions and classes, Logic colloquium '78 (Boffa, M., van Dalen, D., and McAloon, K., editors), North-Holland, Amsterdam, 1979, pp. 159–224.Google Scholar
[6]Feferman, S., Monotone inductive definitions, The L. E. J. Brouwer centenary symposium (Troelstra, A. S. and van Dalen, D., editors), North-Holland, 1982, pp. 77–89.Google Scholar
[7]Feferman, S. and Sieg, W., Proof-theoretic equivalences between classical and constructive theories of analysis, Iterated inductive definitions and subsystems of analysis (Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., editors), Lecture Notes in Mathematics, no. 897, Springer Verlag, Berlin, 1981, pp. 78–142.Google Scholar
[8]Glaβ, T., Standardstrukturen für Systeme expliziter Mathematik, Thesis, University of Münster, 1993.Google Scholar
[9]Griffor, E. and Rathjen, M., The strength of some Martin–Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347–385.CrossRefGoogle Scholar
[10]Harrington, L. A. and Kechris, A. S., On monotone versus nonmonotone induction, Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 888–890.CrossRefGoogle Scholar
[11]Harrington, L. A. and Kechris, A. S., Inductive definability, type-written manuscript, 8 pages.Google Scholar
[12]Kechris, A. S. and Moschovakis, Y. N., Recursion in higher types, Handbook of mathematical logic (Barwise, J., editor), North-Holland, 1977, pp. 681–737.CrossRefGoogle Scholar
[13]Kreisel, G., Generalized inductive definitions, in: Stanford Report on the Foundations of Analysis (mimeographed), Ch. III, Stanford, 1963.Google Scholar
[14]Moschovakis, Y. N., Elementary inductions on abstract structures, North-Holland, Amsterdam, 1974.Google Scholar
[15]Rathjen, M., Untersuchungen zu Teilsystemen der Zahlentheorie zweiter Stufe und der Mengenlehre mit einer zwischen − CA und − CA + BI liegenden Beweisstärke, University of Münster, Institute for Mathematical Logic and Foundational Research, Münster, 1989.Google Scholar
[16]Rathjen, M., Proof-theoretic analysis of KPM, Arch. Mathematical Logic, vol. 30 (1991), pp. 377–403.CrossRefGoogle Scholar
[17]Richter, W., Recursively Mahlo ordinals and inductive definitions, Logic colloquium '69 (Gandy, R. O. and Yates, C. E. M., editors), North-Holland, Amsterdam, 1971, pp. 273–288.CrossRefGoogle Scholar
[18]Richter, W. and Aczel, P., Inductive definitions and reflecting properties of admissible ordinals, Generalized recursion theory (Fenstad, J. E. and Hinman, P. G., editors), North-Holland, Amsterdam, 1974, pp. 301–381.Google Scholar
[19]Takahashi, S., Monotone inductive definitions in a constructive theory of functions and classes, Annals of Pure and Applied Logic (1989), pp. 255–279.Google Scholar