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

On 09.11.2024 15:03, Mikko wrote:

On 2024-11-08 16:30:23 +0000, WM said:

>
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.

 All positive rationals quite obviously are in the sequence. Non-positive
rationals are not.
 

Therefore also irrational numbers cannot be there.

 That is equally obvious.
 

Of course this is wrong.

 You may call it wrong but that's the way they are.

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 rational are outside. That implies that Cantor's above sequence does not contain all rationals.

 

It proves that not all rational numbers are countable and in the sequence.

 Calling a truth wrong does not prove anything.
 

Proving that when Cantor is true the real axis has measure 3 proves that Cantor is wrong.
Regards, WM