Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 2024-11-24 14:01:15 +0000, WM said:
 

On 24.11.2024 13:38, Mikko wrote:

On 2024-11-23 08:49:18 +0000, WM said:
>

It is relevant by three reasons:
1) The limit of the sequence f(n) of relative coverings in (0, n] is 1/10, not 1. Therefore the relative covering 1 would contradict analysis.
2) Since for all intervals (0, n] the relative covering is 1/10, the additional blackies must be taken from the nowhere.
3) Since a shifted blacky leaves a white unit interval where it has left, the white must remain such that the whole real axis can never become black.

>
You say that it is relevant but you don't show how that is relevant
to the fact that there is no real number between the intervals (n/2, n/2+1)
that is not a part of at least one of those intervals.

>
Because that has nothing to do with the topic under discussion. See points 1, 2, and 3. They are to be discussed.

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The subject line specifies that the discussion should be about Cantor's
enumeration of the rational numbers.
>
OP specifies that the discussion shall be baout the sequence of
itnrevals

>
That is a mistake. Should read:
[q_n - ε*sqrt(2)/2^n, q_n + ε*sqrt(2)/2^n].

 OK but the following applies to that, too:
 

The 1, 2, and 3 above are not relevant to the topic sepcified by the
subject line and OP.

 

My last example contradicts a simpler bijection, namely that between all natural numbers and all natural numbers divisible by 10: Let every unit interval on the real axis after a number 10n carry a black hat. Then it should be possible to cover all intervals with black hats.

 What does "contradicts" in "contradicts a simpler bijection"?