Re: Replacement of Cardinality

Am 16.08.2024 um 00:59 schrieb Chris M. Thomasson:

On 8/15/2024 10:06 AM, Python wrote:

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⌋
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NUF(INVNUF(1)) > 1
Contradiction.

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Well, no, this just isn't a proof, sorry about that, Jim.
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(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".)
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Proof by contradiction:
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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!
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(Of course, it's clear that I'm using the same "proof idea" as you used in your attempt of a proof.)

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

DING!!!! Yup! wow. ;^o

Of course I know (believe for good reasons) that I am 100% right. Otherwise I wouldn't post such diatribes. :-P