A consequence of the axiom of choice
- 1 May 1975
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of the Australian Mathematical Society
- Vol. 19 (3) , 306-308
- https://doi.org/10.1017/s1446788700031505
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)). C ≅ R.Keywords
This publication has 1 reference indexed in Scilit:
- A Model of Set-Theory in Which Every Set of Reals is Lebesgue MeasurableAnnals of Mathematics, 1970