Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T17:57:15.708Z Has data issue: false hasContentIssue false

DEDUCTIVE CARDINALITY RESULTS AND NUISANCE-LIKE PRINCIPLES

Published online by Cambridge University Press:  20 April 2021

SEAN C. EBELS-DUGGAN*
Affiliation:
DEPARTMENT OF PHILOSOPHY NORTHWESTERN UNIVERSITYEVANSTON, IL60208, USAE-mail:[email protected]

Abstract

The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof.

Type
Research Article
Copyright
© Association for Symbolic Logic, 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Antonelli, G. A. (2010). Notions of invariance for abstraction principles. Philosophia Mathematica, 18(3), 276292.CrossRefGoogle Scholar
Arana, A., & Detlefsen, M. (2011). Purity of methods. Philosopher’s Imprint, 11(2), 120,Google Scholar
Beaney, M. (1997). The Frege Reader. Oxford, UK: Blackwell.Google Scholar
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 521.CrossRefGoogle Scholar
Boolos, G. (1997). Constructing Cantorian counterexamples. Journal of Philosophical Logic, 26, 237239.CrossRefGoogle Scholar
Boolos, G. (1997). Is Hume’s principle analytic? In Heck, R. K., editors. Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford, UK: Oxford University Press, pp. 245261.Google Scholar
Cook, R. T., editor. (2007). The Arché Papers on the Mathematics of Abstraction, Vol. 71, The Western Ontario Series in Philosophy of Science. Berlin, Germany: Springer.Google Scholar
Cook, R. T., (2009). Diagonalization, the liar paradox, and the inconsistency of the formal system presented in the appendix to Frege’s Grundgesetze: Volume II. In Hieke, A., and Leitgeb, H., editors. Reduction, Abstraction, Analysis, Proceedings of the 31 th International Ludwig Wittgentstein Symposium in Kirchberg, 2008. Boston, MA and Berlin, Germany: De Gruyter. pp. 273288.Google Scholar
Cook, R. T., (2012). Conservativeness, stability, and abstraction. British Journal for the Philosophy of Science, 63, 673696.CrossRefGoogle Scholar
Creed, P., & Truss, J. K. (2001). On quasi-amorphous sets. Archive for Mathematical Logic, 40, 581596.CrossRefGoogle Scholar
Dedekind, R. (1963). Essays on the Theory of Numbers. Mineola, NY: Dover Publications Inc.Google Scholar
Ebels-Duggan, S. C. (2015). The Nuisance principle in infinite settings. Thought: A Journal of Philosophy, 4(4), 263268.Google Scholar
Ebels-Duggan, S. C. (2019). Abstraction principles and the classification of second-order equivalence relations. Notre Dame Journal of Formal Logic, 60(1), 77117.CrossRefGoogle Scholar
Fine, K. (2002). The Limits of Abstraction. Oxford, UK: Clarendon Press.Google Scholar
Frege, G. (1964). The Basic Laws of Arithmetic: Exposition of the System. Berkeley, CA: University of California Press. Translated by Furth, M.CrossRefGoogle Scholar
Frege, G. (1980). The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number (second edition). Evanston, IL: Northwestern University Press, Translated by Austin, J. L.Google Scholar
Frege, G. (2013). Basic Laws of Arithmetic. Oxford, UK: Oxford University Press. Translated by Ebert, P. A., and Rossberg, M.Google Scholar
Grattan-Guinness, I. (1978). How Bertrand Russell discovered his paradox. Historia Mathematica, 5, 127137.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2001). The Reason’s Proper Study. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Heck, R. K. (1992). On the consistency of second-order contextual definitions. Noûs, 26(4), 491494.CrossRefGoogle Scholar
Heck, R. K. (1997). Finitude and Hume’s principle. Journal of Philosophical Logic, 26(6), 589617.CrossRefGoogle Scholar
Kanamori, A. (1997). The mathematical import of Zermelo’s well-ordering theorem. Bulletin of Symbolic Logic, 3, 281311.CrossRefGoogle Scholar
Kanamori, A., & Pincus, D. (2002). Does GCH imply AC locally? In Halász, G., Lovász, L., Simonivits, M., & Sós, V. T., editors. Paul Erdös and his mathematics, volume 2 of Bolyai Society Mathematical Studies, Berlin, Germany: Springer, pp. 413426.Google Scholar
Linnebo, Ø. (2010). Some criteria for acceptable abstraction. Notre Dame Journal of Formal Logic, 52(3), 331338.Google Scholar
Potter, M. (2002). Reason’s Nearest Kin: Philosophies of Arithmetic from Kant to Carnap. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Russell, B. (1906). On some difficulties in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society, 4(14), 2953. Page references are to the reprint in [27].Google Scholar
Russell, B. (1913). The Principles of Mathematics. Cambridge, MA: Cambridge University Press.Google Scholar
Russell, B. (1973). Essays in Analysis. London, UK: George Allen.Google Scholar
Shapiro, S. (1991). Foundations without Foundationalism: A Case for Second-Order Logic, volume 17 of Oxford Logic Guides. New York, NY: The Clarendon Press.Google Scholar
Shapiro, S. (2000). Frege meets Dedekind: A neo-logicist treatment of real analysis. Notre Dame Journal of Formal Logic, 41(4), 335364. Page references are to the reprint in [7].CrossRefGoogle Scholar
Shapiro, S., & Weir, A. (1999). New V, ZF, and abstraction. Philosophia Mathematica, 7, 293321, Reprinted in [7].CrossRefGoogle Scholar
van Heijenoort, J., ed. (1967). From Frege to Gödel : A Source Book in Mathematical Logic. Cambridge, MA: Harvard University Press.Google Scholar
van Mill, J. (1984). An introduction to βω. In Kunen, K., and Vaughan, J. E., editors, Handbook of Set-Theoretic Topology, Amsterdam, Netherlands: North-Holland, pp. 503567.CrossRefGoogle Scholar
Weaver, N. (2014). Forcing for Mathematicians. Singapore: World Scientific Publishing Company.CrossRefGoogle Scholar
Weir, A. (2003). Neo-Fregeanism: An embarassment of riches. Notre Dame Journal of Formal Logic, 44(1), 1348, Reprinted in [7].CrossRefGoogle Scholar
Wright, C. (1997). On the philosophical significance of Frege’s theorem. In Heck, R. K., editor, Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford, UK: Oxford University Press, pp. 201244. Reprinted in [19].Google Scholar
Wright, C. (1998). Is Hume’s principle analytic? Notre Dame Journal of Formal Logic . 40(1), 630. Reprinted in [19] and [7].Google Scholar
Zermelo, E. (1904). A proof that every set can be well-ordered. In van Heijenoort, J., editor. From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge, MA: Harvard University Press, pp. 139141.Google Scholar