Axiom of Choice and Continuum Hypothesis

  • Erwin Engeler


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.


Formal Framework Continuum Hypothesis Mathematical System Theory Uncountable Cardinal Precision Tool 
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Further Reading

  1. Hausdorff, F.: Grundzüge der Mengenlehre, (1914) pp. 399–403, 469–473, New York, Chelsea, (reprint 1949)Google Scholar
  2. Robinson, R.M.: On the Decomposition of Spheres, Fundamenta Mathematicae, vol. 34, PP. 246–260, (1947)MathSciNetMATHGoogle Scholar
  3. Gödel, K.: What is Cantor’s Continuum Problem?, in: P. Benacerraf and H. Putnam: Philosophy of Mathematics, pp. 258–273, Englewood Cliffs, Prentice-Hall, 1964Google Scholar
  4. Gödel, K.: Consistency Proof for the Generalized Continuum-Hypothesis, Proc. Nat. Acad. Sci. USA, vol. 25, pp. 220–224, (1939)CrossRefGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. Scott, D.: A Proof of the Independence of the Continuum Hypothesis, Mathematical System Theory, vol. 1, pp. 89–111, (1967)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Erwin Engeler
    • 1
  1. 1.MathematikdepartmentETH-ZentrumZürichSwitzerland

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