On 17 Jun., 04:37, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > "WM" <mueck...@rz.fh-augsburg.de> 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 countable > > set. > > As I said, all subsets of N are countable. Also as I said, big deal.
No. You misunderstood. Please read carefully. The set of all subsets of N that can be specified somehow is countable.
> > 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. > > Yes.
And so do all subsets of N that can be specified. > > > 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.
But from this list he concludes that the set of reals that can appear as a diagonal, hence can be specified somehow, is uncountable. This conclusion is wrong > > 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.
I understand that aleph_0 is considered to be a number. Then we can conclude that even sets that cannot be enumerated, can have cardinal number aleph_0. Because they are a subset of a set that can be enumerated. > > > > >> ____________________ > >> 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.
The following list contains all reals (i.e. their finite names) that can be specified. This contradicts your claim.
1.1.a 0 2.1.a 1 3.1.a 00 ...
Every line can be expanded to a (countable) infinity of an infinity of lines to cover every word in every language and every alphabet. There is no name missing. The list is counatble.
> > > 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.
Cantor concludes about sets. That is wrong. Further there is a list given above that contains all constructable reals - though some more names. But it would be wrong to assume that a subset has larger cardinal number than its superset. > > > 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.
Why then do they claim that the set of reals is uncountable and conclude that 2^aleph_0 is larger than aleph_0? In fact, that is the error.