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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Tue, 6 Aug 2024 22:36:10 -0400
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On 8/6/24 4:35 AM, WM wrote:
> Le 06/08/2024 à 00:35, Jim Burns a écrit :
>> On 8/5/2024 3:21 PM, WM wrote:
>>> Le 05/08/2024 à 02:24, Moebius a écrit :
>>
>>>> ∃^ℵ₀ u ∈ ⅟ℕ: ∀x > 0: u < x   (false) .
>>>
>>> ∃^NUF(x) u ∈ ⅟ℕ: ∀x > 0: u < x true.
>>
>> ¬∃u ∈ ⅟ℕ: ∀x > 0: u < x
> 
> Right. But with NUF(x) = 1 ==> INVNUF(1) = x we get
> 
> ∃u ∈ ⅟ℕ, u < x, ∀y > x = INVNUF(1).
> 
> Regards, WM
> 
> 

But INVNUF(1) can't exist, as it will be bigger than

1/ ( ceil(1/INVNUF(1)) +1 ), and 1/ ( ceil(1/INVNUF(1)) +2 )
which are two different unit fractions.

And thus proving that it can't exist.