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.
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References
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.
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.
Boolos, G. (1998). Reading the Begriffsschrift. In Jeffrey, R. (Ed.), Logic, logic, and logic (pp. 155–170). Cambridge: Harvard University Press.
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.
Burgess, J. P. (2005). Fixing Frege. Princeton: Princeton University Press.
Burgess, J. P., & Hazen, A. (1998). Arithmetic and predicative logic. Notre Dame Journal of Formal Logic, 39, 1–17.
Dedekind, R. (1963). The nature and meaning of numbers. In Essays on the theory of numbers. New York: Dover. Tr by W. W. Beman.
Ferreira, F. (2005). Amending Frege’s Grundgesetze der Arithmetik. Synthese, 147, 3–19.
Frege, G. (1966). Grundgesetze der arithmetik. Hildeshiem: Georg Olms Verlagsbuchhandlung.
Hajék, P., & Pudlák, P. (1993). Metamathematics of first-order arithmetic. New York: Springer-Verlag.
Hale, B., & Wright, C. (2001). The reason’s proper study. Oxford: Clarendon.
Heck, R. G. (1993). The development of arithmetic in Frege’s Grundgesetze der Arithmetik. Journal of Symbolic Logic, 58, 579–601.
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.
Heck, R. G. (1997). Finitude and Hume’s principle. Journal of Philosophical Logic, 26, 589–617.
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.
Heck, R. G. (2000). Counting, cardinality, and equinumerosity. Notre Dame Journal of Formal Logic, 41, 187–209.
Linnebo, Ø. (2004). Predicative fragments of Frege arithmetic. Bulletin of Symbolic Logic, 10, 153–174.
Wright, C. (1983). Frege’s conception of numbers as objects. Aberdeen: Aberdeen University Press.
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Heck, R.G. Ramified Frege Arithmetic. J Philos Logic 40, 715–735 (2011). https://doi.org/10.1007/s10992-010-9158-y
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DOI: https://doi.org/10.1007/s10992-010-9158-y