[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)
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