FOM: The logical, the set-theoretical, and the mathematical

[Robert M. Solovay]

On Tue, 12 Sep 2000, Kanovei wrote:

> >The Axiom of Choice has a special status.  It is not necessary for the
> >development of number theory, but is certainly an essential part of
> >ordinary mathematical practice for analysis 
> 
> If one commits to consider only Borel objects then all 
> usual instances of Choice necessarily e.g. to prove that 
> a ctble union of null sets is null, become provable in ZF 
> without choice. Yet I don't know if anybody has developed 
> this observation into a careful theory. 
> 

	There is a model due to Azriel Levy of ZF in which the reals are
the countable union of countable sets. This seems to me to directly
contradict the second paragraph of Kanovei's posting.

	--Bob Solovay