Some results on cut-elimination, provable well-orderings, induction and reflection

Annals of Pure and Applied Logic

Volumn 95, Issue 1-3, 1998, Pages 93-184

  Arai, Toshiyasu ^a  

^a HIROSHIMA UNIVERSITY   (Japan)

Author keywords

Cut elimination; Fixed points; Induction; Reflection principles

Indexed keywords

EID: 0032487712     PISSN: 01680072     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-0072(98)00020-7     Document Type: Article

References (55)

1. [1] T. Arai, Variations on a theme by Weiermann, J. Symbol Logic, to appear.
2. [2] H.P. Barendregt, The Lambda Calculus Its Syntax and Semantics, rev. ed., North-Holland, Amsterdam, 1984.
3. [3] A. Beckmann, Separating fragments of bounded arithmetic, PhD thesis, Westfälische Wilhelms-Universität Münster, 1996.
4. [4] M. Beeson, Foundations of Constructive Mathematics. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 6, Springer, Berlin, 1985.
5. [5] L.D. Beklemishev, Induction rules, reflection principles, and provably recursive functions, Ann. Pure Appl. Logic 85 (1997) 193-242.
6. [6] W. Buchholz, Proof-theoretic analysis of termination proofs, Ann. Pure Appl. Logic 75 (1995) 57-65.
7. [7] W. Buchholz, An intuitionistic fixed point theory, Arch. Math. Logic 37 (1997) 21-27.
8. [8] W. Buchholz, S. Feferman, W. Pohlers, W. Sieg, Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies, Lecture Notes in Mathematics, vol. 897, Springer, Berlin, 1981.
9. [9] S. Buss, Bounded Arithmetic, Bibliopolis, Napoli, 1986.
10. [10] S. Buss, J. Krajíček, An application of boolean complexity to separation problems in bounded arithmetic, Proc. London Math. Soc. 69 (1994) 1-21.
11. [11] E.A. Cichon, Termination orderings and complexity characterizations, in: P. Aczel et al. (Eds.), Proof Theory, Cambridge University Press, Cambridge, 1993, pp. 171-193.
12. [12] E.A. Cichon, A. Weiermann, Term rewriting theory for the primitive recursive functions, Ann. Pure Appl. Logic 83 (1997) 199-223.
13. [13] P. Clote, G. Takeuti, First order bounded arithmetic and small boolean circuit complexity classes, in: P. Clote, J.B. Remmel (Eds.), Feasible Mathematics II, Birkhäuser, Boston, 1995, pp. 154-218.
14. [14] N. Dershowitz, J.-P. Jouannaud, Rewrite systems, in: J. van Leeuwen (Ed.), Handbook of Theoretical Computer Science, vol. B, North-Holland, Amsterdam, 1990. pp. 243-320.
15. [15] S. Feferman, Iterated inductive fixed-point theories: Applications to Hancock's conjecture, in: G. Metakides (Ed.), Patras Logic Symp., North-Holland, Amsterdam, 1982, pp. 171-196.
16. [16] H. Friedman, Provably equality in primitive recursive arithmetic with and without induction, Pacific J. Math. 57 (1975) 379-392.
17. [17] H. Friedman, M. Sheard, Elementary descent recursion and proof theory, Ann. Pure Appl. Logic 71 (1995) 1-45.
18. [18] G. Gentzen, Beweibarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie, Math. Ann. 119 (1943) 140-161.
19. [19] J.-Y. Girard, Proof-theoretic investigations of inductive definitions (Part I), in: E. Engeler et al. (Eds.), Logic and Algorithmic, an Internat. Symp. held in Honour of Ernst Specker, Enseign Math. Univ Geneve, Geneve, 1982, pp. 207-236.
20. [20] J.-Y. Girard, Proof Theory and Logical Complexity, vol. 1, Bibliopolis, Napoli, 1987.
21. [21] N. Goodman, Relativized realizability in intuitionistic arithmetic of all finite types, J. Symbol. Logic 43 (1978) 23-44.
22. [22] D. Hofbauer, Termination proofs by multiset path orderings imply primitive recursive derivation lengths, Proc. 2nd, ALP, Lecture Notes in Computer Science, vol. 463, Springer, Berlin, 1990, pp. 347-358.
23. [23] W.A. Howard, G. Kreisel, Transfinite induction and bar induction of type zero and one, and the role of continuity in intuitionistic analysis, J. Symbol Logic 31 (1966) 325-358.
24. [24] J. Hudelmaier, Bounds for cut elimination in intuitionistic prepositional logic, Arch. Math. Logic 31 (1992) 331-353.
25. [25] G. Jäger, R. Kahle, A. Setzer, T. Strahm, The proof-theoretic analysis of transfinitely iterated fixed point theories, submitted.
26. [26] G. Jäger, B. Primo, About the proof-theoretic ordinals of weak fixed point theories, J. Symbol Logic 57 (1992) 1108-1119.
27. [27] G. Jäger, T. Strahm, Fixed point theories and dependent choice, submitted.
28. [28] J. Johannsen, A note on sharply bounded arithmetic, Arch. Math. Logic 33 (1994) 159-165.
29. [29] J. Krajíček, Bounded arithmetic, propositional logic, and complexity theory, Cambridge Univ. Press, Cambridge, 1995.
30. [30] G. Kreisel, A variant to Hubert's theory of foundations of arithmetic, Brit. J. Phil. Sc. 4 (1953) 107-127.
31. [31] G. Kreisel, The status of the first ε-number in first order arithmetic, J. Symbol Logic 25 (1960) 390 (abstract).
32. [32] G. Kreisel, A. Lévy, Reflection principles and their use for establishing the complexity of axiomatic systems, Z. Math. Logik Grundlagen Math. 14 (1968) 97-142.
33. [33] G.E. Mints, Finite investigations of transfinite derivations, in: Selected Papers in Proof Theory, Bibliopolis, Napoli, 1992, pp. 17-72.
34. [34] G.E. Mints, Reflection and transfinite induction, in: Selected Papers in Proof Theory, Bibliopolis, Napoli, 1992, pp. 153-156.
35. [35] Y.N. Moschovakis, Elementary Induction on Abstract Structures, North-Holland, Amsterdam, 1974.
36. [36] W. Pohlers, A short course in ordinal analysis, in: P. Aczel et al. (Eds.), Proof Theory, Cambridge University Press, Cambridge, 1993, pp. 27-78.
37. [37] C. Rüede, T. Strahm, Transfinitely iterated fixed point theories with intuitionistic logic, in preparation.
38. [38] U. Schmerl, A fine structure generated by reflection formulas over primitive recursive arithmetic, in: M. Boffa et al. (Eds.), Logic Colloquium 78, North-Holland, Amsterdam, 1979, pp. 335-350.
39. [39] K. Schütte, Proof Theory, Springer, Berlin, 1977.
40. [40] H. Schwichtenberg, Proof theory: some applications of cut-elimination, in: J. Barwise (Ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 867-897.
41. [41] J.C. Shepherdson, Non-standard models for fragments of number theory, in: J.W. Addison et al. (Eds.), The Theory of Models, North-Holland, Amsterdam, 1965, pp. 342-358.
42. 1 transfinite induction, in: D. van Dalen et al. (Eds.), Logic Colloquium 80, North-Holland, Amsterdam, 1982, pp. 239-253.
43. [43] C. Smorynski, The incompleteness theorems, in: J. Barwise (Ed.), Handbook of Mathematical Logic North-Holland, Amsterdam, 1977, pp. 821-865.
44. [44] T. Strahm, Autonomous fixed point processes and transfinite fixed point recursion, in preparation.
45. [45] R. Sommer, Transfinite inducton within Peano arithmetic, Ann. Pure Appl. Logic 76 (1995) 231-289.
46. [46] G. Takeuti, A remark on Gentzen's paper "Beweibarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie", Proc. Japan Acad. 39 (1963) 263-269.
47. [47] G. Takeuti, Proof Theory, 2nd ed., North-Holland, Amsterdam, 1987.
48. [48] G. Takeuti, RSUV isomorphism, in: P. Clote, J. Krajíček (Eds.), Arithmetic, Proof Theory and Computational Complexity, Oxford Univ. Press, Oxford, 1993, pp. 364-386.
49. [49] G. Takeuti, Separations of theories in weak bounded arithmeic, Ann. Pure Appl. Logic 71 (1995) 47-67.
50. [50] A.S. Troelstra, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973.
51. [51] A.S. Troelstra, D. van Dalen, Constructivism in Mathematics. An Introduction, vol. I, North-Holland, Amsterdam, 1988.
52. [52] S. Wainer, L. Wallen, Basic proof theory, in: P. Aczel et al. (Eds.), Proof Theory, Cambridge University Press, Cambridge, 1993, pp. 1-26.
53. [53] A. Weiermann, Bounding derivation lengths with functions from the slow growing hierarchy, preprint, Münster, 1993.
54. [54] A. Weiermann, Termination proofs by lexicographic path orderings imply multiply recursive derivation lengths, Theoret. Comput. Sci. 139 (1995) 355-362.
55. [55] A. Weiermann, Bounding derivation lengths with functions from the slow growing hierarchy, submitted.