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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: how
Date: Mon, 22 Apr 2024 19:12:39 -0400
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On 4/22/24 10:25 AM, WM wrote:
> Le 20/04/2024 à 20:49, Richard Damon a écrit :
>> On 4/20/24 2:40 PM, WM wrote:
> 
>>>> Of course you can double all the elements, they just double into a 
>>>> subset of themselves.
>>>
>>> Impossible.
>>>
>> But True.
>>
>> It may seem impossible to you, 
> 
> The original set is in (0, ω). The doubled set cannot be compressed to 
> (0, ω). This disproves your unquestioning faith.
> 
> Regards, WM

Why not?

any element of the set (0, ω) will be a finite number 0 < k < ω (since ω 
is DEFINED to be the first transfinite ordinal), and we have as a proven 
fact in mathematics that any finite number, when doubled, will be 
another finite number which will be < ω, and thus the final set IS 
contained in the same bound as the first.

This can also be seen to be due to the fact that aleph_0 * 2 - aleph_0 
(Cardinal arithmetic on Transfinite numbers working differently than 
Ordinal arithmatic).