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

[Kanovei]

> Date: Tue, 12 Sep 2000 23:13:48 -0700 (PDT)
> From: "Robert M. Solovay" <solovay at math.berkeley.edu>
> 
> Sigh.
> 
> 	Levy's model shows one needs choice to prove that the countable
> union of null sets is null.
> 
> 	--Bob Solovay
> 


This is true but not relevant to what I wrote about. 
The relevant observation is that the counterexample, say, 
a countable sequence {X_n} of countable sets X_n whose union is R, 
is not "Borel" in that effective sense which I indicated, e.g., 
because it is not codable by a real.

Vladimir Kanovei


> 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 
> > 
> 
>