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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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 wolfgang.mueckenheim@hs-augsburg.de Posts: 3,394 Registered: 10/18/08
Re: When has countability been separted from listability?
Posted: Oct 5, 2017 6:46 AM

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.

Cantor did. At his times there was no magic "simultaneity" on the one hand and constructivism on the other.
>
> Do you grasp the difference??

I grasp that matheologians will defend their nonsense by the silliest "arguments".

Do you grasp that the sequence 1/1, 1/2, 2/1, ... has been constructed. Cantor even gave the prescription how to construct it.

Abzählbar (countable) is not a technical term for him but has its literal meaning: "Einordnung aller algebraischen Zahlen in Reihenform" (Cantor)

Same with Zermelo's axiom of choice. He proved: Every set can be well-ordered. That means a well-ordering can be done, i.e., constructed.

Regards, WM