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A consequence of the axiom of choice

  • C. J. Ash (a1)
Abstract

Let R, C be the additive groups of the real, complex numbers respectively. Using the Axiom of Choice (A.C.), these groups may be shown to be isomorphic. We show that this cannot be proved in Zermelo-Fraenkel set theory (see e.g. Fraenkel, Bar-Hillel and Levy (1973)) without the additional assumption of A.C. This is one of the most “concrete” used of the Axiom of Choice of which I know. THEOREM 1 (assuming (A.C)). CR.

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Copyright
References
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Derrick J. and Drake F. R. (1967), Independence of the Axiom of Choice from variants of the Continuum Hypothesis. Sets, models and recursion theory, 7584. (North-Holland, Amsterdam).
Fraenkel A. A., Bar-Hillel Y. and Levy A. (1973), Foundations of set theory, 2nd rev, edit. (North- Holland, Amsterdam).
Halmos R. P. (1970), Measure theory. (Van Nostrand, New York, 1970)
Solovay R. A. (1970), Annals of Maths. II 92, 156.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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