Date: Oct 5, 2017 8:16 AM
Author: Alan Smaill
Subject: Re: When has countability been separted from listability?
WM <wolfgang.mueckenheim@hs-augsburg.de> writes:

> 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.

Nostalgic for 1900, I see.

However you are living in the 21st century now.

It take little intelligence to grasp that there are

two logically distinct notions in play here.

Winter understands that.

You muddy the waters as ever.

>> Do you grasp the difference??

>

> I grasp that matheologians will defend their nonsense by the silliest

> "arguments".

So, you do not understand the difference.

>

> Regards, WM

--

Alan Smaill