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

Kanovei kanovei at
Wed Sep 13 01:56:10 EDT 2000

> Date: Tue, 12 Sep 2000 14:55:24 -0700 (PDT)
> From: "Robert M. Solovay" <solovay at>
> 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 

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