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Re: UNCOUNTABILITY
Posted:
Dec 22, 2012 9:55 PM
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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:
>
> > Note, that
> > subcountable
> > does not mean countable.
>
> I am not at all sure of what you mean by subcountable.
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.
So the fact that a set is uncountable need not mean it is "bigger"
than
the natural numbers.
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