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

On 14.01.2025 19:41, Jim Burns wrote:

On 1/14/2025 4:07 AM, WM wrote:

On 13.01.2025 20:31, Jim Burns wrote:

On 1/13/2025 12:17 PM, WM wrote:

 

[...]

>
A step is never from finite to infinite.

The dark realm is appears infinite.

There is no infinite set smaller than ℕ
#𝔼 ≥ #ℕ

>
That is obviously wrong.

 The rule of subset proves that every proper subset has fewer elements than its superset. So there are more natural numbers than prime numbers, || > ||, and more complex numbers than real numbers, || > ||. Even finitely many exceptions from the subset-relation are admitted for infinite subsets. Therefore there are more odd numbers than prime numbers || > ||.
 The rule of construction yields the number of integers || = 2|| + 1 and the number of fractions || = 2||2 + 1 (there are fewer rational numbers # ). Since all products of rational numbers with an irrational number are irrational, there are many more irrational numbers than rational numbers || > |#|.
 The rule of symmetry yields precisely the same number of real geometric points  in every interval (n, n+1] and with at most a small error same number of odd numbers and of even numbers in every finite interval and in the whole real line.

There is no infinite set smaller than ℕ.

{2,3, 4, ...} is smaller by one element.
Regards, WM