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

On 11.11.2024 12:15, Mikko wrote:

On 2024-11-10 10:54:02 +0000, WM said:
 

 

 The measure of all intervals J(n) = [n - √2/10, n + √2/10] is smaller than 3.

 Maybe, maybe not, depending on what is all n.

 It is, as usual, all natural numbers.

 The measure of the interval J(n) is √2/5, which is roghly 0,28.

 Agreed, I said smaller than 3.
 

The measure of the set of all those intervals is infinite.

 The density or relative measure is √2/5. By shifting intervals this density cannot grow. Therefore the intervals cannot cover the real axis, let alone infinitely often.
 

Between the intervals J(n) and (Jn+1) there are infinitely many rational
and irrational numbers but no hatural numbers.

Therefore infinitely many natural numbers must become centres of intervals, if Cantor was right. But that is impossible.