FOM: The logical, the set-theoretical, and the mathematical
Kanovei
kanovei at wmwap1.math.uni-wuppertal.de
Wed Sep 13 05:21:58 EDT 2000
> 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
> >
>
>
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