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On 08.11.2024 14:09, Mikko wrote:All positive rationals quite obviously are in the sequence. Non-positiveOn 2024-11-07 13:21:42 +0000, WM said:If Cantors enumeration of the rationals is complete, then all rationals
On 07.11.2024 10:22, Mikko wrote:Real axis contains both real and irrational numbers and nothing else.On 2024-11-06 17:55:15 +0000, WM said:But not between irrational points.
On 06.11.2024 16:04, Mikko wrote:It is not the measure of the real axis but the set of rationals. TheOn 2024-11-06 10:01:21 +0000, WM said: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.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.
real axis more than just the rationals. The irrationals are also a
part of the real axis.
Between any two points of the real axis there are both rational and
irrational points.
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.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.
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