In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 14 Dez., 22:13, Virgil <vir...@ligriv.com> wrote:
>> Note that the very definition of countability requires that a set >> can be declared countable ONLY if one can demonstrate the existence >> of a surjection from the set of naturals to that set or an injection from that set to the set of naturals.
> If that were correct, there was probably no contradiction. At least it > was not as easy to see. But it is not correct. We have another measure > for countability, namely: every subset of a countable set is > countable.
It may be a "measure", whatever that means, and validly establish countability of subsets of a countable set, or uncountability of supersets of an uncountable set, but it is not the definition of countability.
And showing that a set is a subset of a set that has been shown to be countable shows indirectly that the required surjection or injection must exist.