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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sun, 20 Oct 2024 12:38:35 -0700
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On 10/16/2024 9:55 AM, WM wrote:
> 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.

{ 1*2, 2*2, 3*2, 4*2, 5*2, 6*2, ... }



>>
>> 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
> 
>