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Re: Matheology § 210
Posted:
Feb 7, 2013 2:17 PM
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On Feb 7, 7:59 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 7 Feb., 19:50, William Hughes <wpihug...@gmail.com> wrote:
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> > 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?
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> > Yes. A subset must be constructable.-
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> Sorry, we are in classical set theory. There nothing must be
> constructable.
In classical set theory the accessible numbers are listable
Note from the Wikipedia quote
> Constructively it is consistent to assert the
> subcountability of some uncountable collections
(Wikipedia)
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