Re: Incompleteness of Cantor's enumeration of the rational numbers

On 06.11.2024 21:20, Jim Burns wrote:

On 11/6/2024 2:50 PM, WM wrote:

From every positive point we know that
it is not 0 and
not in contact with (-oo, 0].
Same for every point not in an interval.

 Is 0 "not in contact with" [-1,0) ⊆ ℝ

0 is not a positive point.

A point can be not.in the closure of each
of infinitely.many sets
and also in the closure of their union.

These ideas are irrelevant because we can use the following estimation that should convince everyone:
Use the intervals I(n) = [n - sqrt(2)/2^n, n + sqrt(2)/2^n]. Since n and q_n can be in bijection, these intervals are sufficient to cover all q_n. That means by clever reordering them you can cover the whole positive axis except "boundaries".
And an even more suggestive approximation:
Replace the I(n) by intervals J(n) = [n - 1/10, n + 1/10] (as soon as the I(n) are smaller than 2/10).
These intervals (without splitting or modifying them) can be reordered, to cover the whole positive axis except boundaries.
Reordering them again in an even cleverer way, they can be used to cover the whole positive and negative real axes except boundaries. And reordering them again, they can be used to cover 100 real axes in parallel.
Even using only intervals J(P) = [p - 1/10, p + 1/10] where p is a prime number can accomplish the same.
Is this the power of infinity?
Or is it only the inertia of brains conquered by matheology?
Regards, WM