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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: 5,511 Registered: 9/25/16
Re: When has countability been separted from listability?
Posted: Oct 3, 2017 10:30 AM

Anyway there is no rejoicing that constructible
numbers are enumerable. It still doesn't help
you (directly) to solve equations.

Assume you have an equation:

P(x) = 0

What do you want to do? Enumrate the constructible
numbers x1,x2,... and try them on P(x)=0? Ok if there
is a constructible solution you will find it.

But if there is none, how long will you wait? Thats
why Geometry Theorem proving is not that simple.
You would need something more complete.

Can you factor polynomials, decide it negatively?
https://en.wikipedia.org/wiki/Factorization_of_polynomials

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