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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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Posts: 5,511
Registered: 9/25/16
Re: When has countability been separted from listability?
Posted: Oct 3, 2017 10:22 AM
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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 <> writes:

> > Am Montag, 2. Oktober 2017 11:30:12 UTC+2 schrieb Alan Smaill:
> >> WM <> 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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