[FOM] Algebraic closure of Q

[Harvey Friedman]

On 5/11/06 11:06 AM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> The following question was motivated by a recent discussion on
> sci.math.research.
> 
> Hodges showed that ZF doesn't prove the uniqueness (up to isomorphism) of
> an algebraic closure of Q (W. Hodges, "Lauchli's algebraic closure of Q,"
> Math. Proc. Camb. Phil. Soc. 79 (1976), 289-297).
> 
> Question: What's really needed to prove the existence and uniqueness of
> Qbar?  (There might be more than one answer to this question.)
> 
I take it that you are not concerned with existence, which can be proved in
RCA0 (and much weaker).

I think the uniqueness should be equivalent, over a weak fragment of ZF, to

any countable union of finite sets is countable.

Agree?

Harvey Friedman