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Re: Matheology § 210
Posted:
Feb 7, 2013 1:59 PM
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On 7 Feb., 19:50, William Hughes <wpihug...@gmail.com> wrote:
> On Feb 7, 7:28 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > On 7 Feb., 19:14, William Hughes <wpihug...@gmail.com> wrote:
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> > > On Feb 7, 5:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > > On 7 Feb., 15:56, William Hughes <wpihug...@gmail.com> wrote:
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> > > > > On Feb 7, 3:25 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > > > <snip>
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> > > > > >... a subset S of the countable set F of finite words bijects with
> > > > > > the set D of definable numbers
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> > > > by definition.
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> > > > > Nope. Every D corresponds to some finite word.
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> > > > No, D is a set or at least a collection. A definable number is an
> > > > element of D.
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> > > > > However, S,
> > > > > the collection of all the correspondences, may not be a subset
> > > > > of F (subsets must be computable).
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> > > > Need not be a subset. It is sufficient to know that there are not more
> > > > than countably many correspondences,
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> > > There is no set of correspondences thus there is no number
> > > of correspondences. You cannot know anything about
> > > the number of correspondences.-
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> > You are in error again. There is the axiom of power set. For any F,
> > there is P such that D e P if and only if D c F. According to it every
> > subset of the countable set F exists. Will you dispute that the finite
> > definitions of numbers are a subset of F?
>
> Yes. A subset must be constructable.-
Sorry, we are in classical set theory. There nothing must be
constructable.
Perhaps you are interested in definition and domain of application of
"subcountable"?
In constructive mathematics, a collection is subcountable if there
exists a partial surjection from the natural numbers onto it. The name
derives from the intuitive sense that such a collection is "no bigger"
than the counting numbers. The concept is trivial in classical set
theory, where a set is subcountable if and only if it is finite or
countably infinite. Constructively it is consistent to assert the
subcountability of some uncountable collections such as the real
numbers. (Wikipedia)
So in classical set theory, there is no subcountable set other than
finite or countable set. In particular it need not be constructable.
In that case there were no real numbers in uncountable quantity.
Regards, WM
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