[FOM] Are (C,+) and (R,+) isomorphic

Andreas Blass ablass at umich.edu
Tue Feb 21 09:46:21 EST 2006

Miguel Lerma noted that the axiom of choice is used in the usual 
proof that the additive groups of real numbers and of complex numbers are 
isomorphic, and he asked whether this result can be obtained in ZF without 
choice.  The negative answer to this question is given in the following 

Ash, C. J.
A consequence of the axiom of choice.
J. Austral. Math. Soc. 19 (1975), 306--308.

The review of this paper, by Azriel Levy, on MathSciNet reads:

Let * denote the statement that the additive groups of the real and the 
complex numbers are isomorphic. * is provable from the axiom of choice. 
The author shows that * implies that for every translation invariant 
extension of the Lebesgue measure there is a non-measurable set. The 
latter statment is known to be unprovable without the axiom of choice, 
hence * is unprovable in ZF.

Advertisement: I found this paper quite quickly by looking in the book 
"Consequences of the Axiom of Choice" by Paul Howard and Jean Rubin 
(Mathematical Surveys and Monographs 59; American Mathematical Society; 
1998).  This book is an excellent source for answers to questions of the 
form "Can ... be proved without the axiom of choice?  Is ... equivalent to 
a (familiar) weak version of the axiom of choice?"

Andreas Blass

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