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Topic: When has countability been separted from listability?
Replies: 5   Last Post: Oct 5, 2017 6:46 AM

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bursejan@gmail.com

Posts: 4,626
Registered: 9/25/16
Re: When has countability been separted from listability?
Posted: Oct 3, 2017 10:24 AM
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Corr.:

Show me what bit that is toggled, and I retract my claim...

Am Dienstag, 3. Oktober 2017 16:22:54 UTC+2 schrieb burs...@gmail.com:
> Since the constructible numbers are algebraic numbers,
> with polynomial degree 2^n, they are of course enumerable.
>
> Diagnolization does not work for constructible numbers.
> What bit do you want to toggle to get a new number?
>
> This is complete nonsense, to apply diagnoal argument
> to contructible numbers, you cannot apply it to:
> - rational numbers
> - algebraic numbers
>
> And you cannot apply it to:
> - constructible numbers
>
> You would need to have an inverse mapping from an infinite
> path, to such a number. But there is no such inverse mapping.
>
> Cantors proof hinges on the fact, that the diagonal is an
> element again. You do not have such bits for the numbers above.
>
> You me that bit that is toggled, and I retract my claim...
>
> Am Dienstag, 3. Oktober 2017 14:30:09 UTC+2 schrieb Alan Smaill:

> > WM <wolfgang.mueckenheim@hs-augsburg.de> writes:
> >

> > > Am Montag, 2. Oktober 2017 11:30:12 UTC+2 schrieb Alan Smaill:
> > >> WM <wolfgang.mueckenheim@hs-augsburg.de> writes:
> > >>

> > >> > Cantor has shown that the rational numbers are countable by
> > >> > constructing a sequence or list where all rational numbers
> > >> > appear. Dedekind has shown that the algebraic numbers are countable by
> > >> > constructing a sequence or list where all algebraic numbers
> > >> > appear. There was consens that countability and listability are
> > >> > synonymous. This can also be seen from Cantor's collected works
> > >> > (p. 154) and his correspondence with Dedekind (1882).
> > >> >
> > >> > Meanwhile it has turned out that the set of all constructible real
> > >> > numbers is countable but not listable because then the diagonalization
> > >> > would produce another constructible but not listed real number.

> > >>
> > >> Wrong;

> > >
> > > In my opinion correct, but not invented by me. "The constructable
> > > reals are countable but an enumeration can not be constructed
> > > (otherwise the diagonal argument would lead to a real that has been
> > > constructed)." [Dik T. Winter in "Cantor's diagonalization", sci.math
> > > (7 Apr 1997)]

> >
> > Winter is accurate, you are not.
> > Winter requires the enumeration to be *constructed",
> > following the intuitionistic viewpoint. Cantor did not.
> >
> > Do you grasp the difference??
> >
> >

> > > Regards, WM
> > >

> >
> > --
> > Alan Smaill





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