On Tuesday, 11 June 2013 20:43:30 UTC+2, Zeit Geist wrote:
> I think your algorithm fails at the limit stage.
That's the same with enumerating the rationals and with the diagonal of the list. No limits available in all these cases.
If we enumerate the rationals, we do it up to n (since there is no infinite natural number). If we apply the diagonal argument, we do it up to n (since there is no infinite line-number). If we well-order the rational numbers by magnitude, we do it up to n. More is not possible.
> Just take the natural numbers with a point at infinity appended. Everything works out just fine.
No. There shall be aleph_0 *finite* natural numbers.