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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: Thu, 17 Oct 2024 20:22:12 +0200
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On 17.10.2024 00:39, Jim Burns wrote:

> No natural number is
> the first to not.have a natural.number.double.

True.

>>
>>> When doubling all natural numbers
>>> we obtain only natural numbers.
>>
>> That is impossible.
> 
> There is no first natural number from which we obtain
> (by doubling) anything not.a.natural.number.

True.
> 
> The only set of natural numbers with no first
> is the empty set..

No, the set of dark numbers is another set without smallest element.
> 
> There is no ▒▒▒▒▒ natural number from which we obtain
> (by doubling) anything not.a.natural.number.

Correct is: There is no such _definable_ natnumber.

There is a general rule not open to further discussion:
When doubling natural numbers we obtain even 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