Re: how

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).