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From: WM <wolfgang.mueckenheim@tha.de>
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Le 14/04/2024 à 22:39, Jim Burns a écrit :
> On 4/14/2024 3:12 PM, WM wrote:
>> Le 13/04/2024 à 21:16, Jim Burns a écrit :
> 
>>> if  n < ω
>>> then  2⋅n < ω
>>
>> That is impossible because
>> doubling is a linear operation.
> 
> You (WM) have decided that
> ω is like all the numbers n < ω

Cantor has decided, that ω is an ordinal which can be counted and passed 
by counting (Hilbert).
> 
> Whatever it might mean
> to put ω and 1 on the same line,

It is Cantor's number classes. See Transfinity p. 42.
> 
> If n is a number
> different.in.size from its nearest.neighbors,
> then 2⋅n is a number
> different.in.size from its nearest.neighbors.
> 
> If n is a number less than
>   the least.upper.bound of numbers
>   different.in.size from their nearest.neighbors,
> then 2⋅n is a number less than
>   the least.upper.bound of numbers
>   different.in.size from their nearest.neighbors.

That is wrong if all natnumbers are present already such that no further 
natnumbers fits below ω.
> 
> If  n < ω
> then  2⋅n < ω

That is true if not all natnumbers are present, blocking all places for 
finite ordinals.

Regards, WM