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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: how
Date: Tue, 23 Apr 2024 21:50:22 -0400
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On 4/23/24 2:06 PM, WM wrote:
> Le 22/04/2024 à 20:02, FromTheRafters a écrit :
>> WM has brought this to us :
>>> Le 21/04/2024 à 00:44, FromTheRafters a écrit :
>>>> WM wrote :
>>>
>>>>>> What is a gap in the ordinals?
>>>>>
>>>>> It is a not existing ordinal between two ordinals. It is a not 
>>>>> existing natural number next to ω, for instance.
>>>>
>>>> Sets do not have not existing elements.
>>>
>>> What is ω - 1?
>>
>> An absurdity.
> 
> Not more nor less than ω.
> 
> Regards, WM

But if ω - 1 existed, it would be before ω, and thus less than it.

That is the definition of ordinal numbers.

But since it is NOT a "finite" number, as "ω - 1" + 1 would be ω and not 
a finite number, but all finite numbers + 1 give the next finite number, 
and ω is DEFINED as the first transfinite ordinal, it can not exist.