• Gaisi Takeuti
  • Wilson M. Zaring
Part of the Graduate Texts in Mathematics book series (GTM, volume 1)


In 1895 and 1897 Georg Cantor (1845–1918) published his master works on ordinal and cardinal numbers1. Cantor’s theory of ordinal and cardinal numbers was the culmination of three decades of research on number “aggregates”. Beginning with his paper on the denumerability of infinite sets2, published in 1874, Cantor had built a new theory of the infinite. In this theory a collection of objects, even an infinite collection, is conceived of as a single entity.


Cardinal Number Continuum Hypothesis Primitive Notion Arbitrary Collection Georg Cantor 
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  1. 1.
    Beiträge zur Begründung der transfiniten Mengenlehre. (Erster Artikel.) Math. Ann. Vol. 46, 1895, p. 481–512. (Zweiter Artikel) Math. Ann. Vol. 49, 1897, p. 207–246. For an English translation see Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers. Dover Publications, Inc.Google Scholar
  2. 2.
    Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. J. Reine Angew. Math. Vol. 77, 1874, p. 258–262. In this paper Cantor proves that the set of all algebraic numbers is denumerable and that the set of all real numbers is not denumerable.Google Scholar
  3. 3.
    Atti del IV Congresso Internazionale dei Matematici Roma 1909, Vol. 1, p. 182.Google Scholar
  4. 4.
    van Heijenoort, Jean. From Frege to Gödel. Cambridge: Harvard University Press, 1967. p. 5.MATHGoogle Scholar
  5. 5.
    See What is Cantor’s Continuum Problem? by Kurt Gödel in the Amer. Math. Monthly. Vol. 54 (1947), p. 515–525. A revised and expanded version of this paper is also found in Benacerraf, Paul and Putnam, Hilary. Philosophy of Mathematics Selected Readings. Englewood Cliffs, Prentice-Hall, Inc. 1964.Google Scholar
  6. 6.
    Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publications, Inc.Google Scholar
  7. 7.
    Gödel, Kurt: The Consistency of the Continuum Hypothesis. Princeton: Princeton University Press, 1940.MATHGoogle Scholar
  8. 8.
    Cohen, Paul J.: The Independence of the Continuum Hypothesis. Proceedings of the National Academy of Science of the United States of America, Vol. 50, 1963, pp. 1143–1148.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1971

Authors and Affiliations

  • Gaisi Takeuti
    • 1
  • Wilson M. Zaring
    • 1
  1. 1.University of IllinoisUSA

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