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From: Python <python@invalid.org>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Thu, 15 Aug 2024 19:06:15 +0200
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Le 15/08/2024 à 19:01, Moebius a écrit :
> Am 13.08.2024 um 19:02 schrieb Jim Burns:
>> On 8/13/2024 10:21 AM, WM wrote:
> 
>>> [There is] a real [number] x with NUF(x) = 1.
>>
>> INVNUF(1) > ⅟ ⌊⅟INVNUF(1) +1⌋ > ⅟ ⌊⅟INVNUF(1) +2⌋
>>
>> NUF(INVNUF(1)) > 1
>> Contradiction.
> 
> Well, no, this just isn't a proof, sorry about that, Jim.
> 
> (Hint: The term "INVNUF(1)" is not defined. Hence you may not even use 
> it in a proof by contradiction. Actually, you didn't state an assumption 
> in your "proof".)
> 
> Proof by contradiction:
> 
> Assume that there is an x e IR such that NUF(x) = 1. Let x0 e IR such 
> that NUF(x0) = 1. This means that there is exactly one unit fraction u 
> such that u < x0. Let's call this unit fraction u0. Then (by definition) 
> there is a (actually exactly one) natural number n such that u0 = 1/n. 
> Let n0 e IN such that u0 = 1/n0. But then (again by definition) 1/(n0 + 
> 1) is an unit fraction which is smaller than u0 and hence smaller than 
> x0. Hence NUF(x0) > 1. Contradiction!
> 
> (Of course, it's clear that I'm using the same "proof idea" as you used 
> in your attempt of a proof.)

Of course you are right. 100% right. Any marginally decent high school
student could sketch up the very same proof you've posted.

Our problem, as a community, is not proving Wolfgang Mückenheim wrong.

Our problem, as a community, is that this crook is actually teaching
in an academic institution.