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.) > > If Cantor's diagonal proof results in any such a number, then it > proves in effect the uncountability of a countable set. > > ____________________ > 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.
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. Another proof shows: The reals that can be constructed, are countable. So there is a contradiction. What is the reason: The assumption of finished infinity.
There are no infinite sequences of digits other than established by finite formulas as a_n = 1/n or pi or sqrt(2). Cantor's list with infinitely many infinitely long lines containing arbitrary sequences of digits does not exist (other than by finite formulas - but they are countable). In fact nobody has ever seen that chimera. A typical case of mass hysteria. How could mathematicians be taken in by that rubbish?