Strong Normalization Theorem for a Constructive Arithmetic with Definition by Transfinite Recursion and Bar Induction
Log in :: RSS/Atom Feeds
Notre Dame Journal of Formal Logic

Strong Normalization Theorem for a Constructive Arithmetic with Definition by Transfinite Recursion and Bar Induction

Osamu Takaki

Source: Notre Dame J. Formal Logic Volume 38, Number 3 (1997), 350-373.

Abstract

We prove the strong normalization theorem for the natural deduction system for the constructive arithmetic TRDB (the system with Definition by Transfinite Recursion and Bar induction), which was introduced by Yasugi and Hayashi. We also establish the consistency of this system, applying the strong normalization theorem.

Primary Subjects: 03F05
Secondary Subjects: 03F10
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1039700743
Mathematical Reviews number (MathSciNet): MR1624946
Digital Object Identifier: doi:10.1305/ndjfl/1039700743
Zentralblatt MATH identifier: 0937.03067

References

[1] Girard, J. Y., Proof Theory and Logical Complexity, Bibliopolis, Napoli, 1982.
Mathematical Reviews (MathSciNet): MR89a:03113
Zentralblatt MATH: 0635.03052
[2] Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, New York, 1973.
Mathematical Reviews (MathSciNet): MR48:3699
Zentralblatt MATH: 0275.02025
[3] Yasugi, M., ``Hyper-principle and the functional structure of ordinal diagrams,'' Comment. Math. Univ. st. Pauli, vol. 34 (1985), pp. 227--63; vol. 35 (1986), pp. 1--37.
Zentralblatt MATH: 0633.03052
Mathematical Reviews (MathSciNet): MR815789
[4] Yasugi, M., ``The machinery of consistency proofs,'' Annals of Pure and Applied Logic, vol. 44 (1989), pp. 139--52.
Mathematical Reviews (MathSciNet): MR91f:03116
Zentralblatt MATH: 0679.03023
Digital Object Identifier: doi:10.1016/0168-0072(89)90050-X
[5] Yasugi, M., and S. Hayashi, ``A functional system with transfinitely defined types,'' pp. 31--60 in Lecture Notes in Computer Science, vol. 792, Springer-Verlag, New York, 1994.
Mathematical Reviews (MathSciNet): MR96d:03075
Digital Object Identifier: doi:10.1007/BFb0032393
[6] Yasugi, M., and S. Hayashi, ``Interpretations of transfinite recursion and parametric abstraction in types,'' pp. 452--64 in Words, Languages and Combinatorics 2, World Scientific, Kyoto, 1994.
Mathematical Reviews (MathSciNet): MR96k:03131
Zentralblatt MATH: 0874.03069

2008 © Duke University Press