<email@example.com> wrote in message news:firstname.lastname@example.org... > On Tuesday, 11 June 2013 19:18:34 UTC+2, Julio Di Egidio wrote: >> WM wrote: > >> > Therefore it is not possible to enumerate all rational numbers >> > (always infinitely many remain) by all natural numbers (always >> > infinitely many remain) or to traverse the lines of a Cantor list >> > (always infinitely many remain). >> >> It is not possible to do so effectively... > > It is only possible by applying the axiom of infinity. But that axiom > will also manage to well-order the rationals. After each one has been > attached to a natural, we have the set of naturals that obviously can > be re-ordered in any desired way.
Obviously and crucially not in any finite number of steps (you will get the rationals well-ordered by magnitude from, say, the rationals lexicographically ordered). Anyway, I'm already out of my depths here, so I'll leave this to more competent mathematicians.
> (The power set of |N should even include the singleton of the last natural > as an element.)
There is no such thing as the last natural number, and not even a next to last, and not even a next to next to last, and so on. This was already touched in my initial post, the part you have snipped.
P.S. It's a bit of a pain to have to reformat your posts and quotes every time: it would help a lot if you could at least split your paragraphs in short lines using carriage returns.