On 7 Feb., 15:13, William Hughes <wpihug...@gmail.com> wrote: > On Feb 7, 2:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 7 Feb., 09:10, William Hughes <wpihug...@gmail.com> wrote: > > > > On Feb 7, 9:00 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > <snip> > > > > > What does that mean for the set of accessible numbers? > > > > That this potentially infinite set is not listable. > > > Here we stand firm on the grounds of set theory. > > > Once upon a time there used to be a logocal identity: The expression > > "Set X is countable" used to be equivalent to "Set X can be listed". > > You say, X is countable means there are not more X than any > natural number. The standard term for this is "sub-countable". > Standard terminology is that X is countable iff X is listable. > X can be unlistable and sub-countable, so X can be uncountable > and sub-countable. > > There is nothing contradictory about saying the accessible numbers > are uncountable and sub-countable.
In standard set theory that is a contradiction. Since subcountable is countable, we have X is countable and uncountable.