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A consequence of the axiom of choice

Published online by Cambridge University Press:  09 April 2009

C. J. Ash
C. J. Ash
Affiliation:
Monash UniversityClayton, 3168, Australia
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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)). C ≅ R.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 3 , May 1975 , pp. 306 - 308
Copyright
Copyright © Australian Mathematical Society 1975

References

Derrick, J. and Drake, F. R. (1967), Independence of the Axiom of Choice from variants of the Continuum Hypothesis. Sets, models and recursion theory, 75–84. (North-Holland, Amsterdam).Google Scholar
Fraenkel, A. A., Bar-Hillel, Y. and Levy, A. (1973), Foundations of set theory, 2nd rev, edit. (North- Holland, Amsterdam).Google Scholar
Halmos, R. P. (1970), Measure theory. (Van Nostrand, New York, 1970)Google Scholar
Solovay, R. A. (1970), Annals of Maths. II 92, 1–56.CrossRefGoogle Scholar