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Lectures on proof theory

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Proceedings of the Summer School in Logic Leeds, 1967

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Bibliography

  1. Bachmann, H., Transfinite Zahlen, Springer, Berlin (1955)

    Book  MATH  Google Scholar 

  2. Barwise, K. J., Infinitary logic and admissible sets, Dissertation, Stanford University (1967).

    Google Scholar 

  3. Beth, E. W., On Padoa's method in the theory of definition Indag. Math., v. 15 (1953), pp. 330–339.

    Article  MathSciNet  MATH  Google Scholar 

  4. Choodnovsky, G., Some results in theory of infinitely long expressions, (abstract) Proc. 3d Intl. Cong. for Logic, Methodology and Philos. of Science, Amsterdam, 1967 (to appear).

    Google Scholar 

  5. Craig, W., Linear reasoning. A new form of the Herbrand-Gentzen theorem, Journ. Symbolic Logic, v. 22 (1957), pp. 250–268.

    Article  MathSciNet  MATH  Google Scholar 

  6. -, Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory, Journ. Symbolic Logic, v. 22 (1957), pp. 269–285.

    Article  MathSciNet  MATH  Google Scholar 

  7. Feferman, S., Systems of predicative analysis, Journ. Symbolic Logic, v. 29 (1964) pp. 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  8. Feferman, S., Systems of predicative analysis, II: representations of ordinals, Journ. Symbolic Logic (to appear).

    Google Scholar 

  9. Feferman, S., Autonomous transfinite progressions and the extent of predicative mathematics, Proc. 3d Intl. Cong. for Logic, Methodology and Philos. of Science, Amsterdam, 1967 (to appear).

    Google Scholar 

  10. Feferman, S., Persistent and invariant formulas for outer extensions, Compos. Math. (to appear).

    Google Scholar 

  11. Feferman, S. and G. Kreisel, Persistent and invariant formulas relative to theories of higher order, (Research Announcement) Bull. Amer. Math. Soc. v. 72 (1966), pp. 480–485.

    Article  MathSciNet  MATH  Google Scholar 

  12. Fraïssé, R., Une notion de récursivité relative, in Infinitistic Methods (Proc. Symp. founds. maths, Warsaw, 1959) Pergamon, Oxford (1961), pp. 323–328.

    Google Scholar 

  13. Gentzen, G., Untarsuchungen über das logische Schlaessen, Math. Zeitschr. v. 39 (1934) pp. 176–210, 405–431.

    Article  MathSciNet  MATH  Google Scholar 

  14. -, Die Widerspruchsfreiheit der reinen Zahlentheorie, Math. Annalen, v. 112 (1936), pp. 493–565.

    Article  MathSciNet  MATH  Google Scholar 

  15. -, Neue Fassung des Widerspruchsfreiheitsbeweis fur die reine Zahlentheorie, Forsch. zur Logik und zur Grundlegung der exacten Wissenschaften, no. 4 (1938), Leipzig pp. 19–44.

    MATH  Google Scholar 

  16. -, Beweisbarkeit und Unbeweisbarkeit von Anfangsfallen der transfiniten Induktion in der reinen Zahlentheorie, Math. Annalen, v. 119 (1943) pp. 140–161.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gödel, K., Über formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I. Monats. Math., Physik v. 38 (1931) pp. 173–198 (transl. in [vH], pp. 596–616).

    Article  MATH  Google Scholar 

  18. -, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, v. 12 (1958), pp. 280–287.

    Article  MathSciNet  MATH  Google Scholar 

  19. Grzegorczyk, A., A. Mostowski and C. Ryll-Nardzewski, The classical and the ω-complete arithmetic, Journ. Symbolic Logic, v. 23 (1958), pp. 188–206.

    Article  MathSciNet  MATH  Google Scholar 

  20. van Heijenoort, J. (ed.) From Frege to Gödel (A source book in mathematical logic, 1879–1931), Harvard Univ. Press, Cambridge (1967).

    MATH  Google Scholar 

  21. Henkin, L., An extension of the Craig-Lyndon interpolation theorem, Journ. Symbolic Logic, v. 28 (1963), pp. 201–216.

    Article  MathSciNet  MATH  Google Scholar 

  22. Herbrand, J. Recherches sur la théorie de la demonstration. Trav. Soc. Sciences, Lettres de Varsovie, Cl. III no. 33 (1930), 128 pp. (transl. in part in [vH], pp. 525–581).

    Google Scholar 

    Google Scholar 

  23. Hilbert, D. Über das unendlichen, Math. Annalen, v. 95 (1926), pp. 161–190 (transl. in [vH], pp. 367–392).

    Article  MathSciNet  Google Scholar 

  24. Hilbert, D. and P. Bernays, Grundlagen der Mathematik, vol. 2, Springer, Berlin (1939).

    MATH  Google Scholar 

  25. Karp, C., Languages with expressions of infinite length, North-Holland, Amsterdam (1964).

    MATH  Google Scholar 

  26. Keisler, H. J., Theory of models with generalized atomic formulas, Journ. Symbolic Logic, v. 25 (1960), pp. 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  27. -, Ultraproducts and elementary classes, Indag. Math., v. 23 (1961), pp. 477–495.

    Article  MathSciNet  MATH  Google Scholar 

  28. -, Some applications of infinitely long formulas, Journ. Symbolic Logic, v. 30 (1965), pp. 339–349.

    Article  MathSciNet  MATH  Google Scholar 

  29. Kleene, S. C. Introduction to metamathematics, Van Nostrand, Princeton (1952).

    MATH  Google Scholar 

  30. Kochen, S., Completeness of algebraic systems in higher order calculi, in Summaries of talks presented at the summer institute for symbolic logic, Cornell University 1957, 2nd edition, Inst. for Def. Analyses (1960), pp. 370–376.

    Google Scholar 

  31. -, Ultraproducts in the theory of models, Annals Math., v. 74 (1961) pp. 231–261.

    Article  MathSciNet  MATH  Google Scholar 

  32. Kreisel, G., Model-theoretic invariants: applications to recursive and hyperarithmetic operations, in The theory of models (eds. Addison, Henkin, Tarski), North-Holland, Amsterdam (1965), pp. 190–205.

    Google Scholar 

  33. -, Relative recursiveness in metarecursion theory (abstract) Journ. Symbolic Logic, v. 33 (1969), p. 442.

    MathSciNet  Google Scholar 

  34. Kreisel, G., A survey of proof theory, Journ. Symbolic Logic (to appear).

    Google Scholar 

  35. Kreisel, G., So-called consistency proofs by means of transfinite induction (provable ordinals) (abstract for Leeds meeting ASL, 1967), Journ. Symbolic Logic (to appear).

    Google Scholar 

  36. Kreisel, G. and J. L. Krivine, Elements of mathematical logic (model theory), North-Holland, Amsterdam (1967).

    MATH  Google Scholar 

  37. Kreisel, G. and G. Sacks, Metarecursion theory, Journ. Symbolic Logic, v. 30 pp. 318–338.

    Google Scholar 

  38. Kripke, S., Transfinite recursion on admissible ordinals, I, II, (abstracts) Journ. Symbolic Logic v. 29 (1964), pp. 161–162.

    Google Scholar 

  39. Kunen, K., Implicit definability and infinitary languages, Journ. Symbolic Logic (to appear).

    Google Scholar 

  40. Lévy, A., A hierarchy of formulas in set theory, Memoirs Amer. Math. Soc., no. 57 (1965).

    Google Scholar 

  41. Lopez-Escobar, E. G. K., An interpolation theorem for denumerably long formulas, Fund. Math. v. 57 (1965), pp. 253–272.

    MathSciNet  MATH  Google Scholar 

  42. -, Remarks on an infinitary language with constructive formulas, Journ. Symbolic Logic v. 32 (1967), pp. 305–318.

    Article  MathSciNet  MATH  Google Scholar 

  43. Lorenzen, P., Algebraische und logistische Untersuchungen über freie Verbände, Journ. Symbolic Logic, v. 16 (1951) pp. 81–106.

    Article  MathSciNet  MATH  Google Scholar 

  44. Łos, J., On the extending of models, I, Fund. Math. v. 42 (1955), pp. 38–54.

    MathSciNet  MATH  Google Scholar 

  45. Lyndon, R., An interpolation theorem in the predicate calculus, Pacif. Journ. Math., v. 9 (1959), pp. 129–142.

    Article  MathSciNet  MATH  Google Scholar 

  46. -, Properties preserved under homomorphism, Pacif. Journ. Math. v. 9 (1959), pp. 253–272.

    MathSciNet  MATH  Google Scholar 

  47. Maehara, S., On the interpolation theorem of Craig (Japanese), Sûgaku v. 12 (1960/61), pp. 235–237.

    MathSciNet  MATH  Google Scholar 

  48. Makkai, M., Preservation theorems for the logic with denumerable conjunctions and disjunctions, to appear in Journal Symbolic Logic.

    Google Scholar 

  49. Malitz, J. I., Problems in the model theory of infinite languages, Dissertation, University of California at Berkeley (1965).

    Google Scholar 

  50. Mendelson, E., Introduction to mathematical logic, Van Nostrand, Princeton (1964).

    MATH  Google Scholar 

  51. Novikov, P. S., Inconsistencies of certain logical calculi (Russian) in Infinitistic methods (Proc. symp. founds. maths., Warsaw, 1959), Pergamon, Oxford, 1961, pp. 71–74.

    Google Scholar 

  52. Oberschelp, A., Untersuchungen zur mehrsortigen Quantorenlogik, Math. Annalen, v. 145 (1962), pp. 297–333.

    Article  MathSciNet  MATH  Google Scholar 

  53. Platek, R., Foundations of recursion theory, Dissertation, Stanford University (1966).

    Google Scholar 

  54. Prawitz, D., Naturaldeduction. A proof theoretical study. Acta Univ. Stock., Stockholm studies in philosophy, no. 3 (1965).

    Google Scholar 

  55. Robinson, A., Introduction to model theory and to the metamathematics of algebra, North-Holland, Amsterdam (1963).

    MATH  Google Scholar 

  56. Schütte, K., Beweistheorie, Springer, Berlin (1960).

    MATH  Google Scholar 

  57. -, Predicative well-orderings, in Formal systems and recursive functions (eds. Crossley, Dummett), North-Holland, Amsterdam (1963), pp. 279–302.

    Google Scholar 

  58. -, Eine grenze fur die Beweisbarkeit der Transfiniten Induktion in der verzweigten Typenlogik, Arch. math. Logik, Grundl., v. 7 (1965) pp. 45–60.

    Article  MathSciNet  MATH  Google Scholar 

  59. -, Der Interpolationssatz der intuitionistischen Prädikatenlogik, Math. Annalen, v. 148 (1962), pp. 192–200.

    Article  MATH  Google Scholar 

  60. Scott, D., Logic with denumerable long formulas and finite strings of quantifiers, in The theory of models (eds. Addison, Henkin, Tarski), North-Holland, Amsterdam (1965), pp. 32–341.

    Google Scholar 

  61. Scott, D., Interpolation theorems, Proc. 3d Intl. Cong. for Logic, Methodology and Philos. of Science, Amsterdam 1967 (to appear).

    Google Scholar 

  62. Scott, D. and A. Tarski, Extension principles for algebrakolly closed fields, (abstract) Notices Amer. Math. Soc., v. 5 (1958), pp. 778–779.

    Google Scholar 

  63. Shoenfield, J. R., Mathematical Logic, Addison-Wesley, Reading (1967).

    MATH  Google Scholar 

  64. Smullyan, R. M., A unifying principle in quantification theory, in The theory of models (eds. Addison, Henkin, Tarski), North-Holland, Amsterdam, 1965, pp. 433–434.

    Google Scholar 

  65. Tait, W. W., Normal derivability in classical logic, Journ. Symbolic Logic (to appear).

    Google Scholar 

  66. Takeuti, G., Consistency proofs of subsystems of classical analysis, Annals Math., v. 86 (1967), pp. 299–348.

    Article  MathSciNet  MATH  Google Scholar 

  67. Takeuti, G., Π 11 -comprehension axioms and ω-rule, Proc. Leeds 1967 Inst. in Logic (this volume).

    Google Scholar 

  68. Tarski, A. Contributions to the theory of models, Indag. Math. v. 16 (1954), Part I pp. 572–581, Part II pp. 582–588.

    Article  MathSciNet  MATH  Google Scholar 

  69. Tarski, A. and R. L. Vaught, Arithmetical extensions of relational systems, Compos. Math. v. 13 (1957), pp. 81–102.

    MathSciNet  MATH  Google Scholar 

  70. Vaught, R. L., Applications of the Löwenheim-Skolem-Tarski theorem to problems of completeness and decidability. Indag. Math., v. 16 (1954), pp. 467–472.

    Article  MathSciNet  MATH  Google Scholar 

  71. Wang, H., Logic of many sorted theories, Journ. Symbolic Logic, v. 17 (1952), pp. 105–116.

    Article  MathSciNet  MATH  Google Scholar 

  72. Weil, A., Foundations of algebraic geometry, Amer. Math. Soc. Colloq. Publs. v. 29, Rev. Ed. (1962).

    Google Scholar 

  73. Karp, C., Finite-quantifier equivalence in The theory of models (eds. Addison, Henkin, Tarski), North-Holland, Amsterdam, 1965, pp. 407–412.

    Google Scholar 

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M. H. Löb

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Feferman, S. (1968). Lectures on proof theory. In: Löb, M.H. (eds) Proceedings of the Summer School in Logic Leeds, 1967. Lecture Notes in Mathematics, vol 70. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0079094

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