"Tim Little" <email@example.com> wrote in message news:firstname.lastname@example.org... > On 2010-06-17, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: >> "Virgil" <Virgil@home.esc> wrote in message >>> An infinite set is defined to bee countable if and only if there is a >>> surjection from the set of natural numbers to that set. When such a >>> function is a bijection, it is commonly called a list. >> >> Only if the bijection can be explicitly created. > > You apparently have some bizarre private definition of "list". > > Explicitness has nothing to do with it. > > Though even if it did, you are incorrect. Given any numbering of > Turing machines, a list of computable reals ordered by the > least-numbered Turing machines that compute them is quite explicit. > > The practical difficulties of establishing which Turing machines halt, > which are equivalent and so on are just that: practical difficulties > which have nothing to do with mathematical theory. >
There are a countable number of computable Reals.
You can apply the Cantor construction to any purported list of all computable Reals to form a computable Real not on the list.
This proves that the computable Reals cannot be listed. It does *not* prove the computable Reals are uncountable, and in fact they are not.
In exactly the same manner, Cantor proved that the Reals cannot be listed. This does *not* automatically mean they are uncountable, any more than the same proof applied to computable Reals proves they are uncountable. These are different concepts. (Although they were not when Cantor produced his proof).