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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?
Date: Mon, 7 Oct 2024 10:47:25 +0200
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On 06.10.2024 18:48, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> On 06.10.2024 15:59, Alan Mackenzie wrote:
>>> WM <wolfgang.mueckenheim@tha.de> wrote:
> 
>>>> All unit fractions are separate points on
>>>> the positive real axis, but there are infinitely many for every x > 0.
>>>> That can only hold for definable x, not for all.
> 
>>> Poppycock!  You'll have to do better than that to provide such a
>>> contradiction.
> 
>> It is good enough, but you can't understand.
> 
> I do understand.  I understand that what you are writing is not maths.
> I'm trying to explain to you why.  I've already proved that there are no
> "undefinable" natural numbers.  So assertions about them can not make any
> sense.

You have not understood that all unit fractions are separate points on 
the positive axis. Every point is a singleton set and could be seen as 
such, but it cannot. Hence it is dark.
> 
>>>   Hint: Skilled mathematicians have worked on trying to
>>> prove the inconsistency of maths, without success.
> 
>> What shall that prove? Try to understand.
> 
> It shows that any such results are vanishingly unlikely to be found by
> non-specialists such as you and I.

Unlikely is not impossible.
>> Try only to understand my argument. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0. How can
>> infinitely many unit fractions appear before every x > 0?
> 
> You are getting confused with quantifiers, here.  For each such x, there
> is an infinite set of fractions less than x.  For different x's that set
> varies.  There is no such infinite set which appears before every x > 0.

The set varies but infinitely many elements remain the same. A shrinking 
infinite set which remains infinite has an infinite core.

Regards, WM