Induction over the Continuum

  • Iraj Kalantari
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 9)

Abstract

We exploit the analogy between the well ordering principle for nonempty subsets of N (the set of natural numbers) and the existence of a greatest lower bound for non-empty subsets of [a,b)1 to formulate a principle of induction over the continuum for [a,b) analogous to induction over N. While the gist of the idea for this principle has been alluded to, our formulation seems novel. To demonstrate the efficiency of the approach, we use the new induction form to give a proof of the compactness of [a,b]. (Compactness, which plays a key role in topology, will be briefly discussed.) Although the proof is not fundamentally different from many familiar ones, it is direct and transparent. We also give other applications of the new principle.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Borel, E. (1895). “Sur Quelques Points de la Théorie des Fonctions”, Annales Scientifiques de l’École Normale Supérieure, Paris, 3, 1–55.Google Scholar
  2. [2]
    Bruckner, A.M., Bruckner, J.B. and Thomson, B.S. (1997). Real Analysis, Upper Saddle River (New Jersey): Prentice-Hall.Google Scholar
  3. [3]
    Cousin, P. (1895). “Sur les Fonctions de n-Variables Complexes”, Acta Mathematica 19, 1–61.CrossRefGoogle Scholar
  4. [4]
    Duren, Jr., W.L. (1957). “Mathematical Induction in Sets”, American Mathematical Monthly 64, 19–22.CrossRefGoogle Scholar
  5. [5]
    Ford, L.R. (1957). “Interval-Additive Propositions”, American Mathematical Monthly 64, 106–108.CrossRefGoogle Scholar
  6. [6]
    Frèchet, M. (1906). “Sur Quelques Point du Calcul Fonctionnel”, Rendiconti di Palermo 22, 1–74.Google Scholar
  7. [7]
    Heine E. (1872). “Die Elemente der Funktionenlehre”, Crelles Journal 74, 172–188.Google Scholar
  8. [8]
    Lebesgue, H. (1904). Leçons sur l’Intégration, Paris: Gauthier-Villars.Google Scholar
  9. [9]
    Lindelöf, E.L. (1903). “Sur Quelques Points de la Théorie des Ensembles”, Comptes Rendus Hebdomadaire des Seances de l’Academie des Sciences, Paris, 137, 697–700.Google Scholar
  10. [10]
    Moss, R.M.F. and Roberts, G.T. (1968). “A Creeping Lemma”, American M athematical Monthly 75, 649–652.CrossRefGoogle Scholar
  11. [11]
    Royden, H.L. (1968). Real Analysis, New York: MacMillan.Google Scholar
  12. [12]
    Shanahan, P. (1972). “A Unified Proof of Several Basic Theorems of Real Analysis”, American Mathematical Monthly 79, 895–898.CrossRefGoogle Scholar
  13. [13]
    Shanahan, P. (1974). “Addendum to ‘A Unified Proof of Several Basic Theorems of Real Analysis’” , American Mathematical Monthly 81, 890–891.Google Scholar
  14. [14]
    Veblen, O. (1904). “The Heine-Borel Theorem”, Bulletin of the American Mathematical Society 10, 436–439.CrossRefGoogle Scholar
  15. [15]
    Veblen, O. (1905). “Definitions in Terms of Order Alone in the Linear Continuum and in Well-Ordered Sets”, Transactions of the American Mathematical Society 6, 165–171.CrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Iraj Kalantari
    • 1
  1. 1.Department of MathematicsWestern Illinois UniversityMacombU.S.A.

Personalised recommendations