FOM: The logical, the set-theoretical, and the mathematical
Robert M. Solovay
solovay at math.berkeley.edu
Wed Sep 13 02:13:48 EDT 2000
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
>
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