On Wednesday, 12 June 2013 01:09:56 UTC+2, Ralf Bader wrote:
> we can show that we have got a bijection from the positive rationals to the naturals.
If there are all of them. But since each one is finite, you go only to a finite one. You cannot go beyond every finite natural.
> There is nothing WE do "up to n".
There is. Namely every natural is finite. And every index of a_nn and d_n.
> There is no limit involved.
That is correct. You need the axiom of infinity to "show" that your method holds for "all" n. But that is the same with my well-ordering of the rationals. Every well-ordered subset is enumerated by a natural number. There is no limit necessary because there is no limit involved in set theory. It is enough to have the axiom of infinity to "prove" the well-ordering by magnitude for all rationals.