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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Forgotten to answer?
Date: Sun, 26 Jan 2025 15:14:36 +0100
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On 26.01.2025 14:04, FromTheRafters wrote:
> After serious thinking WM wrote :
>> On 26.01.2025 10:08, FromTheRafters wrote:
>>> WM wrote :
>>>> On 25.01.2025 18:03, FromTheRafters wrote:
>>>>> WM pretended :
>>>>>> On 22.01.2025 19:01, Python wrote:
>>>>>>
>>>>>>  > If you have three coins of 2 euros not a single one is 
>>>>>> "necessary" to
>>>>>> pay a 3 euros drink
>>>>>>
>>>>>> This failing analogy has been repeated again an again, first by 
>>>>>> Rennenkampff, because their authors do not understand the 
>>>>>> principle: Cantor's theorem concerns the set of indices or ordinal 
>>>>>> numbers, not a set of sets.
>>>>>
>>>>> Then how are these 'Cantor's Theorem' ordinals contructed?
>>>>
>>>> That can be done in an arbitrary way.
>>>
>>> Arbitrarily constructed ordinals? Tell me more!
>>
>> Which of {a, b}, {b, c}, {c, a} are required  for the union {a, b, c}?
>>
>> Indexing: 1. {a, b}, 2. {b, c}, 3. {c, a}.
>>
>> The first set is not required because
>>
>> {a, b, c} = {a, b} U {b, c} U {c, a} = {b, c} U {c, a}.
>>
>> The second set is the first required one because
>>
>> {a, b, c} = {b, c} U {c, a} =/= {c, a}.
>>
>> Therefore Cantors's theorem supplies the set of ordinals {2, 3}.
>>
>> By the way, every other choice of indices would yield the same 
>> Cantor-set {2, 3}.
> 
> I see set manipulations but no ordinals at all, arbitrary or not.

The ordinals of necessary sets are 2 and 3. The first necessary ordinal 
is 2.

Regards, WM