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Topic:
When has countability been separted from listability?
Replies:
5
Last Post:
Oct 5, 2017 6:46 AM




Re: When has countability been separted from listability?
Posted:
Oct 5, 2017 6:46 AM


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. > > Do you grasp the difference??
I grasp that matheologians will defend their nonsense by the silliest "arguments".
Do you grasp that the sequence 1/1, 1/2, 2/1, ... has been constructed. Cantor even gave the prescription how to construct it.
Abzählbar (countable) is not a technical term for him but has its literal meaning: "Einordnung aller algebraischen Zahlen in Reihenform" (Cantor)
Same with Zermelo's axiom of choice. He proved: Every set can be wellordered. That means a wellordering can be done, i.e., constructed.
Regards, WM



