In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 16 Jun., 13:15, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> > wrote: > > "WM" <mueck...@rz.fh-augsburg.de> wrote in message > > > > news:email@example.com... > > On 16 Jun., 05:37, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > > > > > > > > > > > > > Virgil <Vir...@home.esc> writes: > > > > But until you can determine which of those 10 cases, how can you > > > > compute the number? > > > > > You can't. The number is computable nonetheless, in the sense that there > > > exists an effective procedure for churning out its decimal expansion. > > > > > As noted, computability is a purely extensional notion. Recall the > > > classical recursion theory exercise, which we find, in some form or > > > other, in pretty much any text on the subject: > > > > > Let f : N --> N be a function such that > > > > > f(x) = 0 if Goldbach's conjecture is true, and 1 otherwise. > > > > > Is f computable? > > > > All computable, Turing-computable, nonetheless computable, definable > > (in any useful, i.e., finite language), and by other means > > determinable numbers form a countable set. > > > > ________________________ > > All subsets of N are countable. Big deal. > > All subsets of N that can be specified somehow belong to a counatble > set. > (In a countable model of ZF this must even be the case. Would it be > impossible, then ZF could not be consistent.)
Even in a countable model, there are sets which cannot be counted WITHIN the model.
> > If Cantor's diagonal proof results in any such a number, then it > > proves in effect the uncountability of a countable set.
No, it just demonstrates that countability with such a model and outside of it may be different. > > > > ____________________ > > What absolute crap. It proves no such thing. And in any event the diagonal > > number is easily computable. Just change all "7"s to "8" and everything else > > to "7". Its a few lines of code. > > No, try to think in a rational way. Every real number that can be > identified by definition, computation, and whatever notions else may > be invented by set theorists to shore up their broken walls, all these > numbers belong to a countable set.
But the very counting of such a countable set provides a construction of countably many non-members, so no such countable set can contain it all.. > > Therefore Cantor "proves" the uncountability of a countable set. > > > ____________________ > > No, it proves there is a Real which is not on the list. > > What is the interpretation? The reals that can be constructed, are > uncountable.
It is proof that there is no countable set of all real numbers, since any alleged such set is provably and constructably incomplete.
Similarly, it is proof that there is no countable set of all constructable numbers, since any alleged such set is provably and constructably incomplete.