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

[Thomas Forster]

...from a colleague of mine here.

It's an uncountably categorical theory: given any two of these
things of the same uncountable power, think of them as vector
spaces over Q. They are then vector spaces of the same dimension
over the one vector space, and so are isomorphic - once was has
a basis! This is where AC comes in. Can one do it with anything
strictly weaker than full AC? I know that if every vector space
has a basis then one gets back AC... but we aren't going to
assume that *every* vector space has a basis.

(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)

   Can anyone shed any light..?



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