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The use of abstract language in elementary metamathematics: Some pedagogic examples

Part of the Lecture Notes in Mathematics book series (LNM,volume 453)

Keywords

  • Atomic Formula
  • Proof Theory
  • Derivation Tree
  • Logical Complexity
  • Satisfaction Relation

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References

  1. Ackermann, W., Die Widerspruchsfreiheit der allgemeinen Mengenlehre, Math. Ann. 114 (1937) 305–315.

    CrossRef  MathSciNet  Google Scholar 

  2. Carstengerdes, W., Mehrsortige logische Systeme mit unendlichlangen Formeln, Archiv Math. Logik Grundlagenforsch 14 (1971) 38–53 and 108–126.

    CrossRef  MATH  Google Scholar 

  3. Chang, C.C. and Keisler, J.H., Continuous model theory, Princeton, 1966.

    Google Scholar 

  4. Dreben, B., Andrews, P., and Aandera, S., False lemmas in Herbrand, Bull. AMS 69 (1963) 699–706.

    CrossRef  MATH  Google Scholar 

  5. Feferman, S., Lectures on proof theory, Springer Lecture Notes 70 (1968) 1–108.

    MathSciNet  Google Scholar 

  6. Friedman, H., Iterated inductive definitions and Σ 12 -AC, pp. 435–442 of: Intuitionism and Proof Theory (ed. Myhill et al.) North Holland Publ. Co., 1970.

    Google Scholar 

  7. Gentzen, G., The collected papers of Gerhard Gentzen, ed. M.E. Szabo, Amsterdam 1969; rev. J. of Philosophy 68 (1971) 238–265.

    CrossRef  Google Scholar 

  8. Girard, J.-Y., Three-valued logic and cut-elimination: the actual meaning of Takeuti's conjecture, Fund. Math. (to appear).

    Google Scholar 

  9. Jockusch, C.G. and Soare, R.I., Π 01 -Classes and degrees of theories. Trans. A.M.S. 173 (1972) 33–56; rev. Ƶbl. 262 (1974) 19; no. 02041.

    MathSciNet  MATH  Google Scholar 

  10. Kreisel, G. and Krivine, J.-L., Elements of mathematical logic, second revised printing, North Holland Publ. Co., 1971.

    Google Scholar 

  11. Kreisel, G. and Levy, A., Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift math. Logik Grundlagen 14 (1968) 97–142.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Lopez-Escobar, E.G.K., On an extremely restricted ω-rule, Fund. Math. (to appear).

    Google Scholar 

  13. Mints, G.E., On E-theorems, Zapiski 40 (1974) 110–118.

    MATH  Google Scholar 

  14. MartinLöf, P., Hauptsatz for the intuitionistic theory of iterated inductive definitions, pp. 179–216 of Proc. Second Scand. Logic Symposium (ed. Fenstad) North Holland Publ. Co., 1971.

    Google Scholar 

  15. Mostowski, A., On recursive models of formalized arithmetic, Bull. Acad. Pol. Sc., cl. III, 5 (1957) 705–710.

    MathSciNet  MATH  Google Scholar 

  16. Kreisel, G., Ordinal logics and the characterization of informal concepts of proof, pp. 289–290 in Proc. ICM Edinburgh 1968.

    Google Scholar 

  17. Parsons, C., Transfinite induction in subsystems of number theory (abstract), JSL 38 (1973) 544.

    Google Scholar 

  18. Schütte, K., Syntactical and semantical properties of simple type theory, JSL 25 (1960) 305–326.

    MATH  Google Scholar 

  19. _____, Beweistheorie, Berlin, 1960.

    Google Scholar 

  20. __, On simple type theory with extensionality, pp. 179–184 in: Logic, Methodology and Philosophy of Science III, North Holland Publ. Co., 1968.

    Google Scholar 

  21. Scott, D.S., Algebras of sets binumerable in complete extensions of arithmetic, pp. 117–121 of: Proc. Symp. Pure Math. 5, AMS, 1962.

    Google Scholar 

  22. Shoenfield, J.R., Mathematical logic, Addison-Wesley, 1967.

    Google Scholar 

  23. Kreisel, G., A survey of proof theory, JSL 33 (1968) 321–388.

    MathSciNet  MATH  Google Scholar 

  24. __, A survey of proof theory II, pp. 109–170 of: Proc. Second Scand. Logic Symp. (ed. Fenstad), North Holland Publ. Co., 1971.

    Google Scholar 

  25. Statman, R., Structural Complexity of proofs, Dissertation, Stanford, 1974.

    Google Scholar 

  26. Takahashi, M., Simple type theory of Gentzen style with the inference of extensionality, Proc. Jap. Acad. 44 (1968) 43–45.

    CrossRef  MATH  Google Scholar 

  27. ____, Many valued logics of extended Gentzen style II, JSL 35 (1970) 493–528.

    Google Scholar 

  28. Troelstra, A.S., Note on the fan theorem, JSL (to appear).

    Google Scholar 

  29. Uesu, T., Zermelo's set theory and G*LC, Comment. Math. Univ. St. Pauli 16 (1967) 69–88.

    MathSciNet  MATH  Google Scholar 

  30. ____, Correction to ‘zermelo's set theory and G*LC', ibid. 19 (1970) 47–49.

    MathSciNet  Google Scholar 

  31. Vaught, R.L., Sentences true in all constructive models, JSL 25 (1960) 39–53.

    MathSciNet  MATH  Google Scholar 

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Kreisel, G., Mints, G.E., Simpson, S.G. (1975). The use of abstract language in elementary metamathematics: Some pedagogic examples. In: Parikh, R. (eds) Logic Colloquium. Lecture Notes in Mathematics, vol 453. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064871

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  • DOI: https://doi.org/10.1007/BFb0064871

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