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

[Robert M. Solovay]

Sigh.

	Levy's model shows one needs choice to prove that the countable
union of null sets is null.

	--Bob Solovay

On Wed, 13 Sep 2000, Kanovei wrote:

> 
> > Date: Tue, 12 Sep 2000 14:55:24 -0700 (PDT)
> > From: "Robert M. Solovay" <solovay at math.berkeley.edu>
> > 
> > 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
>  
> The countable sequence of countable sets which union is 
> the whole R is just not a Borel object in that model, 
> with the understanding of "Borel" as admitting a certain 
> construction coded by a countable wellfounded tree, i.e., 
> roughly, Delta^1_1. That this is the same as members of 
> the smallest sigma-algebra needs itself AC (and is wrong 
> in the Levy's model). 
> 
> Vladimir Kanovei 
>