Skip to main content

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

  • Conference paper
  • First Online:
Logic Colloquium

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Article  MathSciNet  Google Scholar 

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

    Article  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.

    Article  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.

    Article  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.

    Article  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.

    Article  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 

Download references

Authors

Editor information

Rohit Parikh

Rights and permissions

Reprints and permissions

Copyright information

© 1975 Springer-Verlag Berlin · Heidelberg

About this paper

Cite this paper

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

Download citation

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

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07155-6

  • Online ISBN: 978-3-540-37483-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics