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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Mon, 7 Oct 2024 12:59:38 -0700
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On 10/7/2024 5:11 AM, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> 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.
> 
> I understand that full well.  I have a reasonable grasp of point set
> topology.  You don't.
> 
>> Every point is a singleton set and could be seen as such, but it
>> cannot. Hence it is dark.
> 
> That's a complete non-sequitur.  In fact, it's gobbledegook.  Points
> aren't sets.  What "it cannot" refers to is more than unclear.  The same
> applies to the "it" in "it is dark".
> 
> "Dark" would appear to be a seventh synonym for (the negative of)
> "defined" ....
> 
>>>>>    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.
> 
> As near impossible as can be without actually being there.  You cannot
> prove the inconsistency of maths: your understanding of the basics is far
> too limited.  You don't understand the infinite; you don't understand
> point set topology; you don't understand basic set theory.  In fact, your
> understanding is so limited, that you have no idea of the extent of your
> ignorance.  If you had graduated in mathematics you would have a better
> idea of all these things.  But you didn't.

I wonder what his "students" think... The poor lot!


> 
>>>> 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.
> 
> That is not true.  There is no element which is in every one of these
> sets.
> 
>> A shrinking infinite set which remains infinite has an infinite core.
> 
> Again, no.  There is no such thing as a "core", here.  Each of these sets
> has an infinitude of elements.  No element is in all of these sets.
> 
>> Regards, WM
>