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Topic: When has countability been separted from listability?
Replies: 2   Last Post: Oct 6, 2017 4:47 AM

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wolfgang.mueckenheim@hs-augsburg.de

Posts: 3,364
Registered: 10/18/08
Re: When has countability been separted from listability?
Posted: Oct 6, 2017 4:47 AM
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Am Donnerstag, 5. Oktober 2017 14:20:05 UTC+2 schrieb Alan Smaill:
> 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.


No. Nostalgic for mathematics.

> However you are living in the 21st century now.

That does not force me to join a gang of fools who try to rape mathematics and to pervert innocent students.
>
> It take little intelligence to grasp that there are
> two logically distinct notions in play here.


It does take as much and exactly same sort of intelligence as to believe in the immaculate conception.
>
> Winter understands that.


You are proselytizing for matheological nonsense. I will not join.


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

>
> So, you do not understand the difference.


Set theory is definitely not hard to understand. Only set theorists are using this as an argument. Probably because they had a hard time to grasp it. Probably because most are rather stupid. Otherwise it is unexplainable how a man can believe that nonsense even after 20 years of thinking over it.

Regards, WM



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