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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: how
Date: Thu, 18 Apr 2024 11:34:52 -0400
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WM has brought this to us :
> Le 18/04/2024 à 00:59, Richard Damon a écrit :
>> On 4/17/24 2:49 PM, WM wrote:
>
>>> The size of ℕ is |ω|. That means ℕ extends on the ordinal line from 0 to 
>>> ω. By multiplying every natural number the extendion is doubled. That is 
>>> mathematics. Every contrary opinion is foolish.
>> 
>> Nope, the size of ℕ, that is |ℕ| is aleph_0. ω is an ORDINAL number
>
> on the ordinal axis beyond all natural numbers with nothing else before it.
> ℵo = |ℕ| = |ω|.
>
>> (showing order), not a cardinal number (showing size). This means that the 
>> first transfinite ordinal above the set ℕ would be the value ω. So yes, as 
>> you say below, there are no "finite" numbers between the set ℕ and the 
>> value ω, but since ℕ has no "highest" member there is no "predecessor" to 
>> ω, just as there is no predecessor to 0 in the Natural Numbers.
>
> Nonsense.
> ω follows upon all natural numbers. There is nothing between them and ω.

No, it follows the finite ordinals as the first transfinite ordinal.