Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-03T22:03:59.574Z Has data issue: false hasContentIssue false
  • English
  • Français

SET SIZE AND THE PART–WHOLE PRINCIPLE

Published online by Cambridge University Press:  20 September 2013

MATTHEW W. PARKER
MATTHEW W. PARKER*
Affiliation:
Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science
*
*CENTRE FOR PHILOSOPHY OF NATURAL AND SOCIAL SCIENCE LONDON SCHOOL OF ECONOMICS AND POLITICAL SCIENCE HOUGHTON STREET, LONDON WC2A 2AE, UK E-mail: m.parker@lse.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Gödel argued that Cantor’s notion of cardinal number was uniquely correct. More recent work has defended alternative “Euclidean”' theories of set size, in which Cantor’s Principle (two sets have the same size if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part–Whole Principle (if A is a proper subset of B then A is smaller than B). Here we see from simple examples, not that Euclidean theories of set size are wrong, nor merely that they are counterintuitive, but that they must be either very weak or in large part arbitrary and misleading. This limits their epistemic usefulness.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 4 , December 2013 , pp. 589 - 612
Copyright
Copyright © Association for Symbolic Logic 2013 

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

Benci, V., & Di Nasso, M. (2003a). Numerosities of labeled sets: A new way of counting. Advances in Mathematics, 173, 50–65.Google Scholar
Benci, V., & Di Nasso, M. (2003b). Alpha-theory: An elementary axiomatics for non-standard analysis. Expositiones Mathematicae, 21, 355–386.Google Scholar
Benci, V., Di Nasso, M., & Forti, M. (2006). An Aristotelean notion of size. Annals of Pure and Applied Logic, 143, 43–53.Google Scholar
Benci, V., Di Nasso, M., & Forti, M. (2007). An Euclidean measure of size for mathematical universes. Logique et Analyse, 50, 43–62.Google Scholar
Benci, V., Horsten, L., & Wenmackers, S. (2013). Non-Archimedean probability. Milan Journal of Mathematics, 81, 121–151.Google Scholar
Blass, A., Di Nasso, M., & Forti, M. (2012). Quasi-selective ultrafilters and asymptotic numerosities. Advances in Mathematics, 231, 1462–1486.Google Scholar
Bolzano, B. (1950). Paradoxes of the Infinite; transl. by Prihonsky, F., London: Routledge.Google Scholar
Bolzano, B. (1973). Theory of Science; Transl. by Terrell, B., edited by Berg, J., Dordrecht, The Netherlands: D. Reidel Publishing.CrossRefGoogle Scholar
Bunn, R. (1977). Quantitative relations between infinite sets. Annals of Science, 34, 177–191.Google Scholar
Di Nasso, M. (2010). Fine asymptotic densities for sets of natural numbers. Proceedings of the American Mathematical Society, 138, 2657–2665.Google Scholar
Di Nasso, M., & Forti, M. (2010). Numerosities of point sets over the real line. Transactions of the American Mathematical Society, 362, 5355–5371.Google Scholar
Galileo (1939). Dialogues Concerning Two New Sciences. Evanston, IL: Northwestern University. Reprinted by Dover 1954.Google Scholar
Gödel, K. (1947). What is Cantor’s continuum problem? American Mathematical Monthly, 54, 515–525.Google Scholar
Gwiazda, J. (2010). Probability, hyperreals, asymptotic density, and God’s lottery. PhilSci Archive. Available from: http://philsci-archive.pitt.edu/id/eprint/5527.Google Scholar
Heath, T. (1956). The Thirteen Books of Euclid’s Elements. New York: Dover.Google Scholar
Katz, F. M. (1981). Sets and their sizes. PhD Thesis, MIT. Available from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi= 10.1.1.28.7026.Google Scholar
Leibniz, G. W. (2001). The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672–1686. Arthur, R. T. W., translator and editor. New Haven, CT: Yale University Press.Google Scholar
Mancosu, P. (2009). Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? Review of Symbolic Logic, 2, 612–646.Google Scholar
Mayberry, J. (2000). The Foundations of Mathematics in the Theory of Sets. Cambridge: Cambridge University Press.Google Scholar
McCall, S., & Armstrong, D. M. (1989). God’s lottery. Analysis, 49, 223–224.Google Scholar
Parker, M. W. (2009). Philosophical method and Galileo’s paradox of infinity. In van Kerkhove, B., editor. New Perspectives on Mathematical Practices. Hackensack, NJ: World Scientific, pp. 76–113.Google Scholar
Pincus, D., & Solovay, R. (1977). Definability of measures and ultrafilters. Journal of Symbolic Logic, 42, 179–190.Google Scholar
Resnik, M. (1997). Mathematics as a Science of Patterns. Oxford: Clarendon Press.Google Scholar
Tannery, P. (1884). Sur l’authenticité des axioms d’Euclide. Bulletin des sciences mathématiques et astronomiques, 2e série, 8, 162–175.Google Scholar
Wenmackers, S., & Horsten, L. (2013). Fair infinite lotteries. Synthese, 190, 37–61.Google Scholar
Williamson, T. (2007). How probable is an infinite series of heads? Analysis, 67, 173–180.CrossRefGoogle Scholar
Wimsatt, W. C. (1981). Robustness, reliability, and overdetermination. In Brewer, M. B., and Collins, B. E., editors. Scientific Inquiry and the Social Sciences. San Francisco, CA: Jossey Bass, pp. 124–163. Reprinted in W. C. Wimsatt, 2007, Re-engineering Philosophy for Limited Beings, pp. 43–71.Google Scholar