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Axiom of Choice and Continuum Hypothesis

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Foundations of Mathematics

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Abstract

What are the real numbers? At least the question has become somewhat clearer since it was posed in Section 1: any satisfactory answer must provide a frame of reference for mathematical activity, in particular for proving theorems in analysis. There seem to be two aspects of this activity that go hand in hand: on the one hand there is what active mathematicians call “intuition” (without, if they are wise, going into its psychological details), or “thinking in concepts”, on the other the mathematical formalism which, with the help of symbolic logic, can be refined into a precision tool.

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Further Reading

  • Hausdorff, F.: Grundzüge der Mengenlehre, (1914) pp. 399–403, 469–473, New York, Chelsea, (reprint 1949)

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  • Robinson, R.M.: On the Decomposition of Spheres, Fundamenta Mathematicae, vol. 34, PP. 246–260, (1947)

    MathSciNet  MATH  Google Scholar 

  • Gödel, K.: What is Cantor’s Continuum Problem?, in: P. Benacerraf and H. Putnam: Philosophy of Mathematics, pp. 258–273, Englewood Cliffs, Prentice-Hall, 1964

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  • Gödel, K.: Consistency Proof for the Generalized Continuum-Hypothesis, Proc. Nat. Acad. Sci. USA, vol. 25, pp. 220–224, (1939)

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  • Cohen, P.: The Independence of the Continuum Hypothesis I,II, Proc. Nat. Acad. Sci. USA, vol. 50 PP. 1143–1148, and vol. 51, pp. 105–110, (1963, 1964)

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  • Scott, D.: A Proof of the Independence of the Continuum Hypothesis, Mathematical System Theory, vol. 1, pp. 89–111, (1967)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematikdepartment, ETH-Zentrum, CH-8092, Zürich, Switzerland

    Erwin Engeler

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  1. Erwin Engeler
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© 1993 Springer-Verlag Berlin Heidelberg

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Engeler, E. (1993). Axiom of Choice and Continuum Hypothesis. In: Foundations of Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78052-3_5

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  • DOI: https://doi.org/10.1007/978-3-642-78052-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-78054-7

  • Online ISBN: 978-3-642-78052-3

  • eBook Packages: Springer Book Archive

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