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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Tue, 24 Dec 2024 11:42:56 +0100
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On 23.12.2024 15:31, Richard Damon wrote:
> On 12/23/24 3:52 AM, WM wrote:
>> On 23.12.2024 01:05, FromTheRafters wrote:
>>> WM brought next idea :
>>>
>>>> The function E(n) decreases from infinity to zero because in set 
>>>> theory ℕ \ {1, 2, 3, ...} =  { } is an accepted formula.
>>>
>>> In what way is it decreasing?
>>
>> One by one. Every endsegment transforms a natural number from content 
>> inside to index outside. That is guaranteed by mathematics:
>> ∀k ∈ ℕ : ∩{E(0), E(1), E(2), ..., E(k+1)} = ∩{E(0), E(1), E(2), ..., 
>> E(k)} \ {k+1}. Note the universal quantifier.

> Except that you don't understand that infinitiy minus 1 is still 
> infinity, and thus did not "decrease".

The sets E(n) decrease. If the sequence (E(n)) could not get empty one 
by one then Cantor could not set up an infinite sequence using all 
indices n of that sequence.

Regards, WM