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Frege's unofficial arithmetic

Published online by Cambridge University Press:  12 March 2014

Agustín Rayo
Agustín Rayo*
Affiliation:
Department of Logic and Metaphysics, University of St Andrews, St Andrews, KY16 9AL, UK, E-mail: [email protected]
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Abstract

I show that any sentence of nth-order (pure or applied) arithmetic can be expressed with no loss of compositionality as a second-order sentence containing no arithmetical vocabulary, and use this result to prove a completeness theorem for applied arithmetic. More specifically. I set forth an enriched second-order language L. a sentence of L (which is true on the intended interpretation of L), and a compositionally recursive transformation Tr defined on formulas of L, and show that they have the following two properties: (a) in a universe with at least ℶn−2 objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L, and (b) for any sentence of is a second-order sentence containing no arithmetical vocabulary, and

Keywords

Second-order logicarithmeticlogicismnominalismFregeHodesBoolos
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 4 , December 2002 , pp. 1623 - 1638
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

Beaney, Michael (editor) [1997], The Frege reader, Blackwell, Oxford.Google Scholar
Benacerraf, Paul [1965], What numbers could not be, The Philosophical Review, vol. 74, pp. 47–73, Reprinted in Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics.CrossRefGoogle Scholar
Benacerraf, Paul and Putnam, Hilary (editors) [1983], Philosophy of mathematics, second ed., Cambridge University Press, Cambridge.Google Scholar
Boolos, George [1984], To be is to be a value of a variable (or to be some values of some variables), The Journal of Philosophy, vol. 81, pp. 430–49, Reprinted in George Boolos, Logic, Logic and Logic.CrossRefGoogle Scholar
Boolos, George [1985a], Nominalist platonism, Philosophical Review, vol. 94, pp. 327–44, Reprinted in George Boolos, Logic, Logic and Logic.CrossRefGoogle Scholar
Boolos, George [1985b], Reading the Begriffsschrift, Mind, vol. 94, pp. 331–334, Reprinted in George Boolos, Logic, Logic and Logic.CrossRefGoogle Scholar
Boolos, George [1998], Logic, logic and logic, Harvard, Cambridge, Massachusetts.Google Scholar
Bostock, David [1979], Logic and arithmetic. Clarendon Press, Oxford.Google Scholar
Burgess, John and Rosen, Gideon [1997], A subject with no object, Oxford University Press, New York.Google Scholar
Frege, Gottlob [1884], Die Grundlagen der Arithmetik, English Translation by Austin, J.L., The Foundations of Arithmetic, Northwestern University Press, Evanston, IL, 1980.Google Scholar
Frege, Gottlob [1893/1903], Grundgesetze der Arithmetik, Vol 1 (1893), Vol 2 (1903). English Translation by Montgomery Furth, The Basic Laws of Arithmetic, University of California Press, Berkeley and Los Angeles, 1964.Google Scholar
Frege, Gottlob [1919], Notes for Ludwig Darmstaedter, Reprinted in Michael Beaney, The Frege Reader.Google Scholar
Hale, Bob [1997], Grundlagen 64, Proceedings of the Aristotelian Society, vol. 97, pp. 243–61, Reprinted in Bob Hale and Crispin Wright, The Reasons Proper Study.Google Scholar
Hale, Bob and Wright, Crispin [2001], The reason's proper study: Essays towards a neo-Fregean philosophy of mathematics, Clarendon Press.CrossRefGoogle Scholar
Heck, Richard (editor) [1997], Language, thought and logic, Clarendon Press, Oxford.CrossRefGoogle Scholar
Hodes, Harold T. [1984], Logicism and the ontological commitments of arithmetic, Journal of Philosophy, vol. 81, no. 3, pp. 123–149.CrossRefGoogle Scholar
Hodes, Harold T. [1990], Where do natural numbers come from?, Synthese, vol. 84, pp. 347–407.CrossRefGoogle Scholar
McGee, Vann [1997], How we learn mathematical language, Philosophical Review, vol. 106. no. 1, pp. 35–68.CrossRefGoogle Scholar
McGee, Vann [2000], Everything, In Gila Sher and Richard Tieszen. Between Logic and Intuition.Google Scholar
Rayo, Agustín [2002], Word and objects, Noûs, vol. 36, no. 3.CrossRefGoogle Scholar
Rayo, Agustín and Yablo, Stephen [2001], Nominalism through de-nominalization, Noûs, vol. 35, no. 1, pp. 74–92.CrossRefGoogle Scholar
Sher, Gila and Tieszen, Richard (editors) [2000], Between logic and intuition, Cambridge University Press, New York and Cambridge.CrossRefGoogle Scholar
Wright, Crispin [1983], Frege's conception of numbers as objects, Aberdeen University Press, Aberdeen.Google Scholar
Wright, Crispin [1997], The philosophical significance of Frege's theorem. In Richard Heck, Language Thought and Logic.Google Scholar