On 20 Jun., 21:55, Virgil <Vir...@home.esc> wrote:
> > The reals in a certain Cantor-list are countable. And if you form the > > anti-diagonal of that list and add it (for instance at first position) > > to the list, this new list is also countable. Again construct the > > anti-diagonal and add it to the list. Continue. The situation remains > > the same for all anti-diagonals you might want to construct. Therefore > > all reals constructed in that way are countable. Nevertheless it is > > impossible to put all of them in one list, because there would be > > another resulting anti-diagonal. > > > Conclusion: It is impossible to obtain a bijection of all these reals > > with N although all "these" reals are countable with no doubt. > > Equal cardinality of two sets requires, by definition, a bijection > between them. > > The naturals are, by definition of COUNTABLE cardinality so that any set > of equally countable cardinality must have a bijection with the > naturals.
But the set of all diagonals constructed according to the scheme given above is countable but cannot be listed. > > Cantor has shown that the set of all reals cannot biject with the > naturals, ergo, the set of all reals is NOT of countable cardinality.
The same proof shows that some countable sets are of not countable cardinality.