Date: Oct 3, 2017 8:26 AM
Author: Alan Smaill
Subject: Re: When has countability been separted from listability?
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