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Re: When has countability been separted from listability?
Posted:
Oct 6, 2017 4:47 AM


Am Donnerstag, 5. Oktober 2017 14:20:05 UTC+2 schrieb Alan Smaill: > WM <wolfgang.mueckenheim@hsaugsburg.de> writes: > > > Am Dienstag, 3. Oktober 2017 14:30:09 UTC+2 schrieb Alan Smaill: > >> WM <wolfgang.mueckenheim@hsaugsburg.de> writes: > >> > >> > Am Montag, 2. Oktober 2017 11:30:12 UTC+2 schrieb Alan Smaill: > >> >> WM <wolfgang.mueckenheim@hsaugsburg.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



