Advertisement

Euclidean Arithmetic: The Finitary Theory of Finite Sets

  • J.P. Mayberry
Chapter
Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 76)

Abstract

There is a central fallacy that underlies all our thinking about the foundations of arithmetic. It is the conviction that the mere description of the natural numbers as the “successors of zero” (i.e., as what you get by starting at 0 and iterating the operation xx + 1) suffices, on its own, to characterise the order and arithmetical properties of those numbers absolutely. This is what leads us to suppose that the dots of ellipsis in

Keywords

Natural Number Linear Ordering Global Function Arithmetical Function Definite Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Dedekind, R. (1893) Was sind und was sollen die Zahlen? Braunschweig: Vieweg, Translated by Wooster W. Berman as The Nature and Meaning of Numbers in Essays in the Theory of Number, New York: Dover, 1963.Google Scholar
  2. Dedekind, R. (new 1971) Letter to Kefferstein. Translated by H. Wang and S. Bauer-Mengelberg in J. van Heijenoort, ed. (1971) From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931, in J. van Heijenoort, Cambridge, MA: Harvard University Press, 1971.Google Scholar
  3. Frege, G. (1974) The Foundations of Arithmetic, Translated by J.L. Austin, Oxford: Basil Blackwell.Google Scholar
  4. Hajek, P. and Pudlak, P. (1993) Metamathematics of First-Order Arithmetic, in the series Perspectives in Mathematical Logic, Berlin: Springer.zbMATHGoogle Scholar
  5. Homolka, V. (1983) A System of Finite Set Theory Equivalent to Elementary Arithmetic, Ph.D Thesis, University of Bristol.Google Scholar
  6. Mayberry, J.P. (2000) The Foundations of Mathematics in the Theory of Sets, Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  7. Newton, I. (1728) Universal Arithmetic: or a Treatise of Arithmetical Composition and Resolution, London. Reprinted in The Mathematical Works of Isaac Newton Volume II (ed. by Whiteside, D. T.), New York: Johnson Reprint Corporation, 1966.Google Scholar
  8. Pettigrew, R. (2008) The Theories of Natural Number Systems and Infinitesimal Analysis in Euclidean Arithmetic, Ph.D. Thesis, University of Bristol.Google Scholar
  9. Popham, S. (1984) Some Results in Finitary Set Theory, Ph.D. Thesis, University of Bristol.Google Scholar
  10. Vopenka, P. (1979) Mathematics in the Alternative Set Theory, Leipzig: Teubner Verlagsgesellschaft.zbMATHGoogle Scholar
  11. Zermelo, E. (1908) Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen 65, 261–281.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Zermelo, E. (1909) Sur les ensembles finis et le principe de l’induction complète, Acta Mathematica 32, 185–193.MathSciNetCrossRefGoogle Scholar
  13. Zermelo, E. (1930) Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae 14, 29–47.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Dept. of PhilosophyUniversity of BristolBristolEngland, UK

Personalised recommendations