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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: Sat, 12 Oct 2024 19:49:23 +0200
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On 10.10.2024 21:54, joes wrote:
> Am Thu, 10 Oct 2024 20:53:07 +0200 schrieb WM:
>> On 10.10.2024 20:45, Alan Mackenzie wrote:
>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>
>>>> If all natnumbers are there and if 2n is greater than n, then the
>>>> doubled numbers do not fit into ℕ.
>>> For any finite n greater than zero, 2n is greater than n.  The same
>>> does not hold for infinite n.
>> There are no infinite n = natural numbers.
> Exactly! There are furthermore no infinite doubles of naturals (2n).

But the doubles are larger. Hence after doubling the set has a smaller 
density and therefore a larger extension on the real line. Hence not all 
natural numbers have been doubled.
> 
>>>>>> Numbers multiplied by 2 do not remain unchanged.
>>>> Either doubling
>>>> creates new natural numbers. Then not all have been doubled. Or all
>>>> have been doubled, then some products fall outside of ℕ.
>>> No.  Not even close.
>> Deplorable. But note that all natural numbers are finite and follow this
>> law: When doubled then 2n > n. If a set of natural numbers is doubled,
>> then the results cover a larger set than before.. 
> Additionally: if n is finite, so is 2n. It cannot go beyond w.

Then there is no complete set. The doubling can be repeated and 
repeated. Always new numbers are created. Potential infinity.

Regards, WM
>