Date: Oct 3, 2017 8:26 AM
Author: Alan Smaill
Subject: Re: When has countability been separted from listability?
WM <firstname.lastname@example.org> writes:
> Am Montag, 2. Oktober 2017 11:30:12 UTC+2 schrieb Alan Smaill:
>> WM <email@example.com> 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.
> 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