Provably recursive functions in fragments of Peano arithmetic

Hiroakira ONO; Noriya KADOTA

doi:10.2969/jmsj/03840721

Hiroakira ONO, Noriya KADOTA

Author information

JOURNAL FREE ACCESS

1986 Volume 38 Issue 4 Pages 721-737

DOI https://doi.org/10.2969/jmsj/03840721

Details

Published: 1986 Received: October 05, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
Correction information
Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) M. Davis, Computability and Unsolvability, McGraw-Hill, New York, 1958.
2) G. Gentzen, Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie, Math. Ann., 119 (1943), 140-161.
3) J. Ketonen and R. Solovay, Rapidly growing Ramsey functions, Ann. of Math., 113 (1981), 267-314.
4) A. Kino, On provably recursive functions and ordinal recursive functions, J. Math. Soc. Japan, 20 (1968), 456-476.
5) S. C. Kleene, Introduction to Metamathematics, Van Nostrand, New York, 1952.
6) G. Kreisel, On the interpretation of non-finitist proofs, J. Symbolic Logic, 16 (1951), 241-267 and 17 (1952), 43-58.
7) J. D. Monk, Mathematical Logic, Springer, New York-Heidelberg-Berlin, 1976.
8) J. Paris, Some independence results for Peano arithmetic, J. Symbolic Logic, 43 (1978), 725-731.
9) J. Paris, A hierarchy of cuts in models of arithmetic, Lecture Notes in Math., 834, Springer, 1981, pp.312-337.
10) J. Paris and L. Harrington, A mathematical incompleteness in Peano arithmetic, Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 1133-1142.
11) J. Paris and L. Kirby, ∑_n-collection schemas in arithmetic, Logic Colloquium 77, North-Holland, Amsterdam, 1978, pp. 199-209.
12) C. Parsons, Ordinal recursion in partial systems of number theory, Notices Amer. Math. Soc., 13 (1966), 857-858.
13) C. Parsons, On a number theoretic choice schema and its relation to induction, Intuitionism and Proof Theory, North-Holland, Amsterdam and London, 1970, pp. 459-473.
14) K. Shirai, A relation between transfinite induction and mathematical induction in elementary number theory, Tsukuba J. Math., 1 (1977), 91-124.
15) G. Takeuti, Proof Theory, North-Holland, Amsterdam, 1975.
16) S. S. Wainer, A classification of the ordinal recursive functions, Arch. Math. Logik Grundlag., 13 (1970), 61-74.
17) S. S. Wainer, Ordinal recursion and a refinement of the extended Grzegorczyk hierarchy, J. Symbolic Logic, 37 (1972), 281-292.

Right : [1] M. Davis, Computability and Unsolvability, McGraw-Hill, New York, 1958.
[2] G. Gentzen, Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie, Math. Ann., 119 (1943), 140-161.
[3] J. Ketonen and R. Solovay, Rapidly growing Ramsey functions, Ann. of Math., 113 (1981), 267-314.
[4] A. Kino, On provably recursive functions and ordinal recursive functions, J. Math. Soc. Japan, 20 (1968), 456-476.
[5] S. C. Kleene, Introduction to Metamathematics, Van Nostrand, New York, 1952.
[6] G. Kreisel, On the interpretation of non-finitist proofs, J. Symbolic Logic, 16 (1951), 241-267 and 17 (1952), 43-58.
[7] J. D. Monk, Mathematical Logic, Springer, New York-Heidelberg-Berlin, 1976.
[8] J. Paris, Some independence results for Peano arithmetic, J. Symbolic Logic, 43 (1978), 725-731.
[9] J. Paris, A hierarchy of cuts in models of arithmetic, Lecture Notes in Math., 834, Springer, 1981, pp.312-337.
[10] J. Paris and L. Harrington, A mathematical incompleteness in Peano arithmetic, Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 1133-1142.
[11] J. Paris and L. Kirby, ∑_n-collection schemas in arithmetic, Logic Colloquium 77, North-Holland, Amsterdam, 1978, pp. 199-209.
[12] C. Parsons, Ordinal recursion in partial systems of number theory, Notices Amer. Math. Soc., 13 (1966), 857-858.
[13] C. Parsons, On a number theoretic choice schema and its relation to induction, Intuitionism and Proof Theory, North-Holland, Amsterdam and London, 1970, pp. 459-473.
[14] K. Shirai, A relation between transfinite induction and mathematical induction in elementary number theory, Tsukuba J. Math., 1 (1977), 91-124.
[15] G. Takeuti, Proof Theory, North-Holland, Amsterdam, 1975.
[16] S. S. Wainer, A classification of the ordinal recursive functions, Arch. Math. Logik Grundlag., 13 (1970), 61-74.
[17] S. S. Wainer, Ordinal recursion and a refinement of the extended Grzegorczyk hierarchy, J. Symbolic Logic, 37 (1972), 281-292.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Corresponding author

Correction information

Register with J-STAGE for free!