In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 3 Jan., 22:36, Virgil <vir...@ligriv.com> wrote: > > > > It is prerequisite when dealing with numbers. And it is exactly what > > > Cantor applied, He distinguished numbers at finite places. > > > > Cantor may have distinguished numerals at finite places, but numerals > > > So Cantor "proved" that there are uncountably many numerals?
Actually, Cantor proved that the set of real numbers was not countable in the sense that no mapping from N to R can be surjective, which is what, by definition, being countable requires.
> Why then does anybody believe in uncountably many numbers?
Because counting is provably unable to exhaust them.
Does WM have some definition of countability of a set other than existence of surjection from N to the set in question?
If so, he should present his alternative ASAP, because as long as that definition holds, the reals remain provably uncountable. --