A normalization theorem in formal theories of natural numbers

Shigeru HINATA

doi:10.2969/jmsj/02920327

Shigeru HINATA

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JOURNAL FREE ACCESS

1977 Volume 29 Issue 2 Pages 327-340

DOI https://doi.org/10.2969/jmsj/02920327

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Published: 1977 Received: February 16, 1976 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) G. Gentzen, Untersuchungen über das logische Schließen, Math. Z., 39 (1935), 176-210, 405-431.
2) G. Gentzen, Neue Fassung des Widerspruchsfreiheitsbeweiss für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, New Series, 4, Leipzig (Hirzel), 1938, 19-44. English translation, in: The collected papers of Gerhard Gentzen, ed. M. E. Szabo, Amsterdam (North-Holland), 1969, 252-286.
3) R. Harrop, Concerning formulas of types A→B∨C, A→(Ex)B(x) in intuitionistic formal systems, J. Symbolic Logic, 25 (1960), 27-32.
4) S. Hinata, Calculability of primitive recursive functionals of finite type, Sci. Rep. Tokyo Kyoiku Daigaku, 9 (1967), 218-235. Errata, J. Symbolic Logic., 13 (1975), 151.
5) H. R. Jervell, A normalform in first order arithmetic, in: Proceedings of the second Scandinavian logic symposium, ed. J. E. Fenstad, Amsterdam (North-Holland), 1971, 93-108.
6) P. Martin-Löf, Hauptsatz for the intuitionistic theory of iterated inductive definitions, ed. J. E. Fenstad, Amsterdam (North-Holland)., 179-216.
7) D. Prawitz, Ideas and results of proof theory, ed. J. E. Fenstad, Amsterdam (North-Holland)., 235-307.
8) B. Scarpellini, Some applications of Gentzens second consistency proof, Math. Ann., 181 (1969), 325-344.
9) B. Scarpellini, On cut elimination in intuitionistic systems of analysis, in: Intuitionism and proof theory, eds. A. Kino, J. Myhill and R. E. Vesley, Amsterdam (North-Holland), 1970, 271-285.
10) B. Scarpellini, Proof theory and intuitionistic systems, Lecture notes in mathematics, 212, Berlin, Heidelberg, New York (Springer), 1971.
11) B. Scarpellini, Induction and transfinite induction in intuitionistic systems, Ann. Math. Logic, 4 (1972), 173-227.
12) G. Takeuti, On the fundamental conjecture of GLC I, J. Math. Soc. Japan, 7 (1955), 249-275.
13) A. S. Troelstra et al., Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture notes in mathematics, 344, Berlin, Heidelberg, New York (Springer), 1973.
14) J. Zucker, Correspondence between cut-elimination and normalization, Ann. Math. Logic, 7 (1974), 1-112, 113-155.

Right : [1] G. Gentzen, Untersuchungen über das logische Schließen, Math. Z., 39 (1935), 176-210, 405-431.
[2] G. Gentzen, Neue Fassung des Widerspruchsfreiheitsbeweiss für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, New Series, 4, Leipzig (Hirzel), 1938, 19-44. English translation, in: The collected papers of Gerhard Gentzen, ed. M. E. Szabo, Amsterdam (North-Holland), 1969, 252-286.
[3] R. Harrop, Concerning formulas of types A→B∨C, A→(Ex)B(x) in intuitionistic formal systems, J. Symbolic Logic, 25 (1960), 27-32.
[4] S. Hinata, Calculability of primitive recursive functionals of finite type, Sci. Rep. Tokyo Kyoiku Daigaku, 9 (1967), 218-235. Errata, Calculability of primitive recursive functionals of finite type, Sci. Rep. Tokyo Kyoiku Daigaku, 13 (1975), 151.
[5] H. R. Jervell, A normalform in first order arithmetic, in: Proceedings of the second Scandinavian logic symposium, ed. J. E. Fenstad, Amsterdam (North-Holland), 1971, 93-108.
[6] P. Martin-Löf, Hauptsatz for the intuitionistic theory of iterated inductive definitions, ed. J. E. Fenstad, Amsterdam (North-Holland), 179-216.
[7] D. Prawitz, Ideas and results of proof theory, ed. J. E. Fenstad, Amsterdam (North-Holland), 235-307.
[8] B. Scarpellini, Some applications of Gentzens second consistency proof, Math. Ann., 181 (1969), 325-344.
[9] B. Scarpellini, On cut elimination in intuitionistic systems of analysis, in: Intuitionism and proof theory, eds. A. Kino, J. Myhill and R. E. Vesley, Amsterdam (North-Holland), 1970, 271-285.
[10] B. Scarpellini, Proof theory and intuitionistic systems, Lecture notes in mathematics, 212, Berlin, Heidelberg, New York (Springer), 1971.
[11] B. Scarpellini, Induction and transfinite induction in intuitionistic systems, Ann. Math. Logic, 4 (1972), 173-227.
[12] G. Takeuti, On the fundamental conjecture of GLC I, J. Math. Soc. Japan, 7 (1955), 249-275.
[13] A. S. Troelstra et al., Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture notes in mathematics, 344, Berlin, Heidelberg, New York (Springer), 1973.
[14] J. Zucker, Correspondence between cut-elimination and normalization, Ann. Math. Logic, 7 (1974), 1-112, 113-155.

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