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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: how
Date: Thu, 18 Apr 2024 18:42:16 -0400
Organization: i2pn2 (i2pn.org)
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On 4/18/24 10:59 AM, WM wrote:
> 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.
> 
> Regards, WM
> 
> 
> 


Right, and any Natural Number * 2 is a Natural Number, so less than ω.

Your logic can't handle the fact that the set of Natural  Numbers is 
unbounded on the high side, so it doesn't understand that.