[FOM] Are (C,+) and (R,+) isomorphic?
Miguel A. Lerma
mlerma at math.northwestern.edu
Mon Feb 20 10:26:06 EST 2006
I am conducting an elementary math problem solving group and,
unexpectedly, the solution to one of the (supposedly "elementary")
problems has led to a question of foundations.
The solution to the problem involves a group-isomorphism between (C,+)
and (R,+), i.e., between the additive groups of complex and real
numbers. But are they really isomorphic? The only proof I have in
mind resorts to the fact that they are Q-vector spaces of the same
dimension (the cardinality of the continuum), so they are isomorphic
as vector spaces over Q (rational numbers), and consequently they are
isomorphic as additive groups.
However that is a highly non-constructive proof, and am not sure
whether it would work without resorting to the Axiom of Choice. So,
this is the question: is there any model of ZF (without AC) in which
(C,+) and (R,+) are not isomorphic?
Miguel A. Lerma
--
Miguel A. Lerma
Math Comp Sys Admin (former Math Lecturer)
NU Math Problem Solving Group Coordinator
Department of Mathematics <mlerma at math.northwestern.edu>
Northwestern University <http://www.math.northwestern.edu/~mlerma/>
2033 Sheridan Road 847-491-8020 (w)
Evanston, IL 60208-2730 847-491-8906 (f)
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