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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 3, 2017 10:24 AM


Corr.:
Show me what bit that is toggled, and I retract my claim...
Am Dienstag, 3. Oktober 2017 16:22:54 UTC+2 schrieb burs...@gmail.com: > Since the constructible numbers are algebraic numbers, > with polynomial degree 2^n, they are of course enumerable. > > Diagnolization does not work for constructible numbers. > What bit do you want to toggle to get a new number? > > This is complete nonsense, to apply diagnoal argument > to contructible numbers, you cannot apply it to: >  rational numbers >  algebraic numbers > > And you cannot apply it to: >  constructible numbers > > You would need to have an inverse mapping from an infinite > path, to such a number. But there is no such inverse mapping. > > Cantors proof hinges on the fact, that the diagonal is an > element again. You do not have such bits for the numbers above. > > You me that bit that is toggled, and I retract my claim... > > 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. > > > > Do you grasp the difference?? > > > > > > > Regards, WM > > > > > > >  > > Alan Smaill



