Journal of Philosophical Logic

, Volume 40, Issue 6, pp 715–735 | Cite as

Ramified Frege Arithmetic

Article

Abstract

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

Keyword

Frege predicativity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boolos, G. (1998). Frege’s theorem and the Peano postulates. In Jeffrey, R. (Ed.), Logic, logic, and logic (pp. 291–300). Cambridge: Harvard University Press.Google Scholar
  2. 2.
    Boolos, G. (1998). On the proof of Frege’s theorem. In Jeffrey, R. (Ed.), Logic, logic, and logic (pp. 275–291). Cambridge: Harvard University Press.Google Scholar
  3. 3.
    Boolos, G. (1998). Reading the Begriffsschrift. In Jeffrey, R. (Ed.), Logic, logic, and logic (pp. 155–170). Cambridge: Harvard University Press.Google Scholar
  4. 4.
    Boolos, G., & Heck, R. G. (1998). Die Grundlagen der Arithmetik §§82–83. In Jeffrey, R. (Ed.), Logic, logic, and logic (pp. 315–338). Cambridge: Harvard University Press.Google Scholar
  5. 5.
    Burgess, J. P. (2005). Fixing Frege. Princeton: Princeton University Press.Google Scholar
  6. 6.
    Burgess, J. P., & Hazen, A. (1998). Arithmetic and predicative logic. Notre Dame Journal of Formal Logic, 39, 1–17.CrossRefGoogle Scholar
  7. 7.
    Dedekind, R. (1963). The nature and meaning of numbers. In Essays on the theory of numbers. New York: Dover. Tr by W. W. Beman.Google Scholar
  8. 8.
    Ferreira, F. (2005). Amending Frege’s Grundgesetze der Arithmetik. Synthese, 147, 3–19.CrossRefGoogle Scholar
  9. 9.
    Frege, G. (1966). Grundgesetze der arithmetik. Hildeshiem: Georg Olms Verlagsbuchhandlung.Google Scholar
  10. 10.
    Hajék, P., & Pudlák, P. (1993). Metamathematics of first-order arithmetic. New York: Springer-Verlag.Google Scholar
  11. 11.
    Hale, B., & Wright, C. (2001). The reason’s proper study. Oxford: Clarendon.Google Scholar
  12. 12.
    Heck, R. G. (1993). The development of arithmetic in Frege’s Grundgesetze der Arithmetik. Journal of Symbolic Logic, 58, 579–601.CrossRefGoogle Scholar
  13. 13.
    Heck, R. G. (1995). Definition by induction in Frege’s Grundgesetze der Arithmetik. In Demopoulos, W. (Ed.), Frege’s philosophy of mathematics (pp. 295–333). Cambridge: Harvard University Press.Google Scholar
  14. 14.
    Heck, R. G. (1997). Finitude and Hume’s principle. Journal of Philosophical Logic, 26, 589–617.CrossRefGoogle Scholar
  15. 15.
    Heck, R. G. (1997). The Julius Caesar objection. In R. Heck (Ed.), Language, thought, and logic: Essays in honour of Michael Dummett (pp. 273–308). Oxford: Clarendon Press.Google Scholar
  16. 16.
    Heck, R. G. (2000). Counting, cardinality, and equinumerosity. Notre Dame Journal of Formal Logic, 41, 187–209.CrossRefGoogle Scholar
  17. 17.
    Linnebo, Ø. (2004). Predicative fragments of Frege arithmetic. Bulletin of Symbolic Logic, 10, 153–174.CrossRefGoogle Scholar
  18. 18.
    Wright, C. (1983). Frege’s conception of numbers as objects. Aberdeen: Aberdeen University Press.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of PhilosophyBrown UniversityProvidenceUSA

Personalised recommendations