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

On 08.11.2024 14:09, Mikko wrote:

On 2024-11-07 13:21:42 +0000, WM said:
 

On 07.11.2024 10:22, Mikko wrote:

On 2024-11-06 17:55:15 +0000, WM said:
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On 06.11.2024 16:04, Mikko wrote:

On 2024-11-06 10:01:21 +0000, WM said:

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I leave ε = 1. No shrinking. Every point outside of the intervals is nearer to an endpoint than to the contents.

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This discussion started with message that clearly discussed limits when
ε approaches 0. The case ε = 1 was only about a specific unimportant
question.

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When ε approaches 0 then the measure of the real axis is, according to Cantor's results, 0. That shows that his results are wrong.

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It is not the measure of the real axis but the set of rationals. The
real axis more than just the rationals. The irrationals are also a
part of the real axis.

>
But not between irrational points.

 Real axis contains both real and irrational numbers and nothing else.
Between any two points of the real axis there are both rational and
irrational points.

If Cantors enumeration of the rationals is complete, then all rationals are in the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1,  ... and none is outside. Therefore also irrational numbers cannot be there. Of course this is wrong. It proves that not all rational numbers are countable and in the sequence.

It is Cantor's result that all rationals are countable, hence inside my intervals.

 That is but what you said above is not.
 

But we can use the following estimation that should convince everyone:
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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".

 Depends on the type of n.

The n are the natural numbers.

 

And an even more suggestive approximation:
Replace the I(n) by intervals J(n) = [n - 1/10, n + 1/10].

 Likewise.

These are the intervals sketched:
 >> J(n) = [n - 1/10, n + 1/10]
 >> --------_1_--------_2_--------_3_--------_4_--------_5_--------_...
Only a very hard believer can believe that by shuffling the intervals they could cover the real line infinitely often.
Regards, WM