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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit fractions?
Date: Mon, 07 Oct 2024 05:38:32 -0400
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WM wrote :
> 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.

Wow, your shrinking sets again? Sets don't change.