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Re: UNCOUNTABILITY
Posted:
Dec 23, 2012 7:50 PM
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On Dec 23, 1:12 am, Virgil <vir...@ligriv.com> wrote:
> In article
> <ee1d96e2-fbb9-4c48-a02e-cef0a3204...@s14g2000yqg.googlegroups.com>,
> William Hughes <wpihug...@gmail.com> wrote:
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> > On Dec 22, 7:04 pm, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <2263ace1-dfa7-466d-8341-c50692402...@v7g2000yqv.googlegroups.com>,
> > > William Hughes <wpihug...@gmail.com> wrote:
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> > > > Note, that
> > > > subcountable
> > > > does not mean countable.
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> > > I am not at all sure of what you mean by subcountable.
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> > A set X is subcountable if we can associate a different natural number
> > with every element x of X, call it f(x) In classical mathematics
> > subcountable
> > implies countable because f(X) must be a subset of the natural
> > numbers.
> > However, if we take a contructivist viewpoint, then we do not know
> > that f(X) is a subset (it may not be contructable). So in
> > constructive
> > mathematics the fact that X is subcountable, does not mean we can
> > find a bijection between X and some subset of the naturals, so X might
> > not be countable. E.g. in constructive mathematics the (constructive)
> > reals
> > are subcountable but not countable.
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> > So the fact that a set is uncountable need not mean it is "bigger"
> > than
> > the natural numbers.
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> But the constraints of your "constructive mathematics' are not required
> in classical mathematics when not doing your constructive mathematics,
> so are not relevant in classical mathematics.
Indeed. However the original post in this thread was concerned
with the affect of definability on Cantor's argument.
I note that Cantor's theorem is perfectly valid with the
assumption that no unconstructable object exists,
there is no (contructable) list of all (contructable) reals,
so the reals remain uncountable. My remarks are aimed
at the obvious question, "If every constructable number is given by
a string, is there not an injection from the constructable numbers
to the naturals, and hence are the constructable numbers
not countable?" The problem is the collection of all naturals
which represent constructable numbers is not a constructable
subset of the naturals.
I am not a constructivist, though I see the appeal.
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