[FOM] A question about uncountable torsion-free divisible groups

[Ashutosh]

On 3/21/12, Thomas Forster <T.Forster at dpmms.cam.ac.uk> wrote:

> (Actually my colleague's specific question was about how much
> choice one needs to prove that $\Re$ and $\Re^2$ are iso as
> abelian groups, but the general question seems to be of interest)

I think every group isomorphism between R and R^2 is non Lebesgue
measurable and does not have the property of Baire. So you need more
than dependent choice.

I think Simon Thomas has asked on Mathoverflow if a non principal
ultrafilter on the set of integers could be used to construct such an
isomorphism. He also asked if, in the presence on a non principal
ultrafilter on integers, the field of complex numbers could have an
automorphism other than identity and conjugation.

Ashutosh