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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Wed, 16 Oct 2024 07:54:42 -0400
Organization: i2pn2 (i2pn.org)
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On 10/16/24 4:30 AM, WM wrote:
> On 15.10.2024 00:27, Chris M. Thomasson wrote:
> 
>> Take the index, using natural numbers
>>
>> 1: { 1 }
>> 2: { 1, 2 }
>> 3: { 1, 2, 3 }
>> 4: { 1, 2, 3, 4 }
>> ...
>>
>> vs:
>>
>> 1: { 2 }
>> 2: { 2, 4 }
>> 3: { 2, 4, 6 }
>> 4: { 2, 4, 6, 8 }
>> ...
>>
> There is a general rule not open to further discussion:
> When doubling natural numbers we obtain natural numbers which have not 
> been doubled.
> 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.
> 
> Regards, WM
> 

But that isn't actually a general rule, so you are just admitting that 
you aren't talking about the same Natural Numbers that everyone else is 
talking about, but some bastardization that is just finite.

NO natural number when doubled results in a value that wasn't a natural 
number all the time.

Your problem is that your "Acutal Infinity" isn't actually Infinity, and 
like isn't actually "actually".