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Predicative Fragments of Frege Arithmetic

Published online by Cambridge University Press:  15 January 2014

Øystein Linnebo
Øystein Linnebo*
Affiliation:
Department of Philosophy, University of Oslo, Postboks 1020 Blindern, N-0315 Oslo, NorwayE-mail: , [email protected]
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Abstract

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume's Principle, which says that the number of Fs is identical to the number of Gs if and only if the Fs and the Gs can be one-to-one correlated. According to Frege's Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume's Principle, the other, with the underlying second-order logic—and investigates how much of Frege's Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 2 , June 2004 , pp. 153 - 174
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[1] Behmann, Heinrich, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem, Mathematische Annalen, vol. 86 (1922), pp. 419–432.Google Scholar
[2] Benacerraf, Paul and Putnam, Hilary (editors), Philosophy of mathematics: Selected readings, Cambridge, Cambridge University Press, 1983, second edition.Google Scholar
[3] Bernays, Paul, Onplatonism in mathematics, 1935, reprinted in [2].Google Scholar
[4] Boolos, George, To be is to be a value of a variable (or to be some values of some variables), Journal of Philosophy, vol. 81 (1984), pp. 430–449, reprinted in [9].Google Scholar
[5] Boolos, George, Nominalist platonism, Philosophical Review, vol. 94 (1985), pp. 327–344, reprinted in [9].Google Scholar
[6] Boolos, George, The consistency of Frege's foundations of arithmetic, On beings and sayings: Essays in honor of Richard Cartwright (Thomson, J.J., editor), MIT Press, Cambridge, MA, 1987, reprinted in [9] and [12].Google Scholar
[7] Boolos, George, The standard of equality of numbers, Meaning and method: Essays in honor of Hilary Putnam (Boolos, George, editor), Harvard University Press, Cambridge, MA, 1990, reprinted in [9] and [12].Google Scholar
[8] Boolos, George, Is Hume's principle analytic?, Logic, language, and thought (Heck, Richard, editor), Oxford University Press, Oxford, 1997, reprinted in [9].Google Scholar
[9] Boolos, George, Logic, logic, and logic, Harvard University Press, Cambridge, MA, 1998.Google Scholar
[10] Burgess, John P., Review of Crispin Wright's: Frege's conception of numbers as objects, Philosophical Review, vol. 93 (1984), pp. 638–40.CrossRefGoogle Scholar
[11] Burgess, John P., Fixing Frege, Princeton University Press, Princeton, NJ, forthcoming.Google Scholar
[12] Demopoulos, William (editor), Frege's philosophy of mathematics, Harvard University Press, Cambridge, MA, 1995.Google Scholar
[13] Dummett, Michael, Neo-Fregeans in bad company?, 1998, in [36].Google Scholar
[14] Feferman, Solomon and Hellman, Geoffrey, Predicative foundations of arithmetic, Journal of Philosophical Logic, vol. 24 (1995), pp. 1–17.CrossRefGoogle Scholar
[15] Feferman, Solomon and Hellman, Geoffrey, Challenges to predicative foundations of arithmetic, Between logic and intuition (Sher, Gila and Tieszen, Richard, editors), Cambridge University Press, Cambridge, 2000.Google Scholar
[16] Fine, Kit, The limits of abstraction, Oxford University Press, Oxford, 2002.Google Scholar
[17] Begriffsschrift, Gottlob Frege, a formula language, modeled upon that of arithmetic, for pure thought, 1879, reprinted in [38].Google Scholar
[18] Begriffsschrift, Gottlob Frege, Foundations of arithmetic, Blackwell, Oxford, 1953, translated by Austin, J.L.. Excerpts reprinted in [2].Google Scholar
[19] Begriffsschrift, Gottlob Frege, Basic laws of arithmetic, University of California Press, Berkeley and Los Angeles, 1964, edited and translated by Furth, Montgomery.Google Scholar
[20] Geach, Peter, Review of M. Dummett, Frege: Philosophy of language, Mind, vol. 84 (1975), pp. 436–499.Google Scholar
[21] Hale, Bob and Wright, Crispin, Reason's proper study, Clarendon, Oxford, 2001.CrossRefGoogle Scholar
[22] Hazen, A.P., Review of Crispin Wright's: Frege's concept of numbers as objects, Australasian Journal of Philosophy, vol. 63 (1985), pp. 251–254.Google Scholar
[23] Heck, Richard G. Jr., The development of arithmetic in Frege's grundgesetze der arithmetik, The Journal of Symbolic Logic, vol. 58 (1993), pp. 579–601, reprinted in [12].Google Scholar
[24] Heck, Richard G. Jr., The consistency of predicative fragments of Frege's grundgesetze der arithmetik, History and Philosophy of Logic, vol. 17 (1996), pp. 209–220.Google Scholar
[25] Heck, Richard G. Jr., Finitude and Hume's principle, Journal of Philosophical Logic, vol. 26 (1997), pp. 598–617.Google Scholar
[26] Heck, Richard G. Jr., The Julius Caesar objection, Language, thought, and logic: Essays in honour of M. Dummett (Heck, Richard G. Jr., editor), Oxford University Press, Oxford, 1997.Google Scholar
[27] Heck, Richard G. Jr., Cardinality counting, and equinumerosity, Notre Dame Journal of Formal Logic, vol. 41 (2000), pp. 187–209.Google Scholar
[28] Lewis, David, Parts of classes, Blackwell, Oxford, 1991.Google Scholar
[29] Linnebo, Øystein, Frege's proof of referentiality, forthcoming in Notre Dame Journal of Formal Logic.Google Scholar
[30] Linnebo, Øystein, Plural quantification exposed, Noûs, vol. 37 (2003), pp. 71–92.Google Scholar
[31] Löwenheim, Leopold, Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76 (1915), pp. 447–470, translated as “On possibilities in the calculus of relatives” in [38].Google Scholar
[32] Parsons, Charles, Frege's theory of number, 1965, reprinted in [12] and [33].Google Scholar
[33] Parsons, Charles, Mathematics in philosophy, Cornell University Press, Ithaca, NY, 1983.Google Scholar
[34] Rumfitt, Ian, Hume's principle and the number of all objects, NoÜs, vol. 35 (2001), pp. 515–41.Google Scholar
[35] Russell, Bertrand, Letter to Frege, 1902, in [38].Google Scholar
[36] Schirn, Matthias (editor), Philosophy of mathematics today, Oxford, Clarendon, 1998.Google Scholar
[37] Tennant, Neil, On the necessary existence of numbers, Noûs, vol. 31 (1997), pp. 307–336.Google Scholar
[38] van Heijenoort, Jean (editor), From Frege to Gödel, Cambridge, MA, Harvard University Press, 1967.Google Scholar
[39] Wright, Crispin, Frege's conception of numbers as objects, Aberdeen University Press, Aberdeen, 1983.Google Scholar
[40] Wright, Crispin, The harmless impredicativity of N= (Hume's principle), 1998, in [36]; reprinted in [21].Google Scholar
[41] Wright, Crispin, Response to Michael Dummett, 1998, in [36]; reprinted in [21].Google Scholar
[42] Zalta, Edward, Natural numbers and natural cardinals as abstract objects: A partial reconstruction of Frege's grundgesetze in object theory , Journal of Philosophical Logic, vol. 28 (1999), pp. 619–660.Google Scholar