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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Wed, 16 Oct 2024 18:55:58 +0200
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On 16.10.2024 17:34, Jim Burns wrote:
> On 10/16/2024 4:50 AM, WM wrote:

>>> There is a general rule not open to further discussion:
>>> When doubling natural numbers
>>> we obtain natural numbers which
>>> have not been doubled.
>>
>> CORRECTION:
>> When doubling natural numbers
>> we obtain even numbers which have not been doubled.
> 
> The set S of ordinals which
> are finite  and for which
> their double is not finite
> is empty.

Maybe. Then not all natural numbers have been doubled.
> 
> When doubling natural numbers (finite ordinals)
> we obtain natural numbers.

Maybe.
> 
> When doubling all natural numbers
> we obtain only natural numbers.

That is impossible.
> 
>>> In potential infinity
>>> we obtain more even natural numbers
>>> than have been doubled.
>>> In actual infinity
>>> we double ℕ and obtain
>>> neither ℕ or a subset of ℕ.
> 
> There is a general rule not open to further discussion:
> The natural numbers ℕ equal the finite ordinals 𝕆ᶠⁱⁿ

That is only possible in potential infinity. But there the result is 
worthless.

Regards, WM