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Explicit mathematics with monotone inductive definitions: A survey

from PART III - APPLICATIVE AND SELF-APPLICATIVE THEORIES

Published online by Cambridge University Press:  31 March 2017

By
Michael Rathjen
Edited by
Wilfried Sieg ,
Richard Sommer  and
Carolyn Talcott
Wilfried Sieg
Affiliation:
Carnegie Mellon University, Pennsylvania
Richard Sommer
Affiliation:
Stanford University, California
Carolyn Talcott
Affiliation:
Stanford University, California
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Reflections on the Foundations of Mathematics
Essays in Honor of Solomon Feferman
 , pp. 329 - 346
Publisher: Cambridge University Press
Print publication year: 2002

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References

[1] W., Buchholz, S., Feferman, W., Pohlers,W., Sieg: Iterated inductive definitions and subsystems of analysis, Lecture Notes inMath. 897 (Springer, Berlin, 1981) 78-142.
[2] D., Cenzer: Ordinal recursion and inductive definitions in: E., Fenstad and P., Hinman: Generalized Recursion Theory I (North-Holland, Amsterdam, 1974) 221-264.
[3] S., Feferman: A language and axioms for explicit mathematics, in: J.N., Crossley (ed.): Algebra and Logic, Lecture Notes inMath. 450 (Springer, Berlin 1975) 87-139.
[4] S., Feferman: Constructive theories of functions and classes in: Boffa, M., van Dalen, D., McAloon, K. (eds.), Logic Colloquium –78 (North-Holland, Amsterdam 1979) 159-224.
[5] S., Feferman: Monotone inductive definitions in: Troelstra, A. S., van Dalen, D.|(eds), The L.E.J. Brouwer Centenary Symposium (North-Holland, Amsterdam, 1982) 77-89.
[6] S., Feferman: Monotone inductive definitions in: Troelstra, A. S., van Dalen, D.|(eds), The L.E.J. Brouwer Centenary Symposium (North-Holland, Amsterdam, 1982) 77-89.
S.|FefermanandW. Sieg: Proof-theoretic Equivalences between classical and constructive theories of analysis, in: W., Buchholz, S., Feferman, W., Pohlers, W., Sieg: Iterated inductive definitions and subsystems of analysis, Lecture Notes inMath. 897 (Springer, Berlin, 1981) 78-142.
[7] T., Glas: Standardstrukturen für Systeme Expliziter Mathematik, Inaugural-Dissertation (Münster, 1993).
[8] T., Glas, M., Rathjen, A., Schlüter: The strength of monotone inductive definitions in explicit mathematics, Annals of Pure and Applied Logic 85 (1997) 1-46.Google Scholar
[9] L.A., Harrington: Kolmogorov's R-operator and the first nonprojectible ordinal, mimeographed notes (1975) 13 pages.
[10] L.A., Harrington and A.S., Kechris: On monotone versus nonmonotone induction, Bull. Am. Math. Soc. 82, 888-890 (1976).Google Scholar
[11] L.A., Harrington and A.S., Kechris: Inductive definability, unpublished, type-written manuscript, 8 pages.
[12] G., Jäger: A well-ordering proof for Feferman's theory T0, Archiv f. Math. Logik 23 (1983) 65-77.
[13] G., Jäger and W., Pohlers: Eine beweistheoretische Untersuchung von Δ12 - CA + BI und verwandter Systeme, Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch- Naturwissenschaftliche Klasse (1982).
[14] A.S., Kechris: Spector second order classes and reflection, in: J.E., Fenstad, R.O., Gandy, G.E., Sacks: Generalized recursion theory II (North-Holland, Amsterdam, 1978) 147-183.
[15] G., Kreisel: Generalized inductive definitions, in: Stanford Report on the Foundations of Analysis (mimeographed), CH. III, Stanford 1963.
[16] T., John: Recursion in Kolmogorov's R-operator and the ordinal _ 3, Journal of Symbolic Logic 51 (1986) 1-11.
[17] M., Rathjen: Untersuchungen zu Teilsystemen der Zahlentheorie zweiter Stufe und der Mengenlehre mit einer zwischen Δ12 - CA und Δ12 - CA + BI liegenden Beweisstärke (Publication of the Institute for Mathematical Logic and Foundational Research of the University ofMünster, 1989).
[18] M., Rathjen: Monotone inductive definitions in explicit mathematics. Journal of Symbolic Logic 61 (1996) 125-146.Google Scholar
[19] M., Rathjen: Explicit mathematics with the monotone fixed point principle. Journal of Symbolic Logic 63 (1998) 509-542.Google Scholar
[20] M., Rathjen: Explicit mathematics with the monotone fixed point principle. II:Models. Journal of Symbolic Logic 64 (1999) 517-550.Google Scholar
[21] S., Takahashi: Monotone inductive definitions in a constructive theory of functions and classes, Ann. Pure Appl. Logic 42 (1989) 255-279.Google Scholar

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