On Tuesday, June 11, 2013 11:57:32 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > 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. >
I'm sorry please disregard,
"You right, no difference. Both fail at the limit stage."
I mis read that and didn't see the part of you post.
I want to examine your "Well Ordering of the Rations Proof" before comment further.
I will say this, though.
When finding a limit ( what happens at the end ), we need to add that limit ordinal, Omega. That like doing a supertask. But I don't find it necessary to do a supertask for most, if not all of these operations.
For example, for Zeno, crossing the room is a physical act, so use Physics. Now, real analysis works nicely in facilitating Physics, and it contains that limit process. Define some functions for the motion and figure out when he crossed the room. We are beyond such ancient "Paradoxes".
Cantors diagonal is not such a thing.
For two real numbers a and b, we have
If there exists n e |N, such that a_n ~= b_n; then a ~= b.
We must show that given a countable list of real numbers: There exists a real number, r, such that r is not on the list. Hence, we must show that: If x e L, the list, then r ~= x.
We find the anti-diagonal suffice for our r.
For then, if x e L and it is the m-th element, we have x_m ~= r_m. And hence, x ~= r, for x e L.
It's a different case from taking the limit of all cases, as in C = |N in the other list.
Here, we don't take limits. We find that x ~= r, for all x e L in one simple steps, not the same step over and over. The limits were done in the developing of the Reals.
> > > 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. >