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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Fri, 27 Dec 2024 22:57:14 +0100
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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.
>>>
>>> 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, ...} .

> .
> .
> .
>