"WM" <email@example.com> wrote in message news:firstname.lastname@example.org... > 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.
As I said, all subsets of N are countable. Also as I said, big deal.
> (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. Cantor's proof is not about a "set" of Reals; it is about a "list" of Reals.
You seem to not understand that something can be a set but cannot be enumerated as a list, which is pretty much the whole point of Cantor's proof.
> >> ____________________ >> 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.
No. The interpretation is that you cannot form a list of all Reals, and any list which purports to be so must miss at least one Real.
> 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. >
No, the pretty obvious (and correct) conclusion is that you cannot form a list of the constructable Reals.
Note, again, that you are implicitly treating a list of Reals as the same thing as a set of Reals; Cantor's proof is about Lists, not Sets.
> 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? >
Well, mathematicians don't confuse lists of Reals with sets of Reals.