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

On 08.11.2024 21:36, Moebius wrote:

Am 08.11.2024 um 21:35 schrieb Moebius:

What is the measure you are using and what does it give for the real
axis?

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Ob Du es nochmal schaffst, auf diesen saudummen Scheißdreck NICHT zu antworten?

 Ich kann es jedenfalls nicht mehr sehen/lesen.

Kein Wunder, weil es Deinen starken Glauben erschüttert.
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.
The measure of all intervals J(n) = [n - √2/10, n + √2/10] is smaller than 3. If no irrationals are outside, then nothing is outside, then the measure of the real axis is smaller than 3. That is wrong. Therefore there are irrationals outside. That implies that rationals are outside. That implies that Cantor's above sequence does not contain all rationals.
And an even more suggestive approximation:
Replace the I(n) by intervals J(n) = [n - 1/10, n + 1/10].
These intervals (without splitting or modifying them) can be reordered, to cover the whole positive axis except boundaries.
But note that Cantor's bijection between naturals and rationals does not insert any non-natural number into ℕ. It confirms only that both sets are very large. Therefore also the above sequence of intervals keeps the same intervals and the same reality. And the same density 1/5 for every finite interval and therefore also in the limit.
Regards, WM