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

Am 26.12.2024 um 05:09 schrieb Moebius:

Am 26.12.2024 um 05:02 schrieb Chris M. Thomasson:

On 12/24/2024 4:07 PM, Ross Finlayson wrote:

On 12/24/2024 12:27 PM, Chris M. Thomasson wrote:

 

Cantor's Pairing works with any unsigned integer.

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No, it works with two copies of all the integers, [...]

>
It works with any unsigned integer.

 It works especially with the (elements in the) two sets {0, 2, 4, ...} and {1, 3, 5, ...}:
                    n <-> n+1 .
 P := {(n, n+1) : n e {0, 2, 4, ...}} .
 Then P is {(0, 1}, (2, 3}, (4, 5}, ...}.

This way we prove that
      {0, 2, 4, ...} ~ {1, 3, 5, ...} ,
and hence
      card {0, 2, 4, ...} = card {1, 3, 5, ...} .

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