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

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

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:
 

On 06.11.2024 16:04, Mikko wrote:

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

 

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

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

 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.

 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.

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.

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

Calling a truth wrong does not prove anything.
--
Mikko