In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> 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.
Whatever do yu mean by "the set of all diagonals"? If you mean one "diagonal" for each possible list, that set of diagonals is not countable. If you mean the set of all "diagonals" for a single list, that is not countable either, since each of uncountably many permutations of the list produces a different "diagonal". > > > > 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.
I have yet to see such a "proof" that was not as flawed ad WM's views on mathematics. > > But you prefer not to think about that?
I don't mind thinking about it at all, but I do not choose to believe anything that requires WM's perverse assumptions about mathematics to justify it.