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

On 03.12.2024 23:32, Moebius wrote:

"Set-theoretical limit" and "coventional limit" (as defined in real analysis) are different notions.

In particular because the set-theoretical limit must be attained by the quantized elements. Every natural number must exist and added to the function in order to complete the set ℕ. (Otherwise no bijection could ever have been claimed.) Therefore dark elements are necessary.

 Maybe it would be helful to write "LIM_(n->oo) ..." (instead of "lim_(n->oo) ...") for the former ...
 

LIM(n-->oo) E(1)∩E(2)∩...∩E(n) = LIM(n-->oo) E(n) = {0}.
The limit is attained step by step:
∀k ∈ ℕ : E(k+1) = E(k) \ {k}
Finite endsegments contain only dark numbers.
Regards, WM