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.
No. X is countable means there is a bijection of X with N.
> The standard term for this is "sub-countable". > Standard terminology is that X is countable iff X is listable.
X is countable if and only if there is a bijection of X with N.
> 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.
There is a contradiction (in mathematics, of course there is no contradiction in matheology because there are no contradictions - it is surprising that even this word con... can be used in matheology) since a subset S of the countable set F of finite words bijects with the set D of definable numbers. F is listable, D is unlistable, i.e., uncountable.