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Subject: Re: Replacement of Cardinality
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Date: Tue, 06 Aug 24 13:52:42 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 06/08/2024 à 12:32, Jim Burns a écrit :
> On 8/6/2024 4:26 AM, WM wrote:
>> Le 06/08/2024 à 00:19, Jim Burns a écrit :

>> NUF(x) gives
>> the number of unit fractions smaller than x.

>> For NUF(x) = 3
>> ⅟ℕᵈᵉᶠ∩(0,x) is finite, namely 3.
> 
> For NUF(x) = 3.5
> ⅟ℕᵈᵉᶠ∩(0,x) is fractional, namely 3.5, however,
> no such x with NUF(x) = 3.5  exists.

That is not of interest. (We could however subdivide the distance between 
u_3 and u_4.)
> 
> Also,
> no such x with NUF(x) = 3  exists.

At least we have found now a way to express finitely many unit fractions 
without the accusation of quantifier shift and without the insane result 
that for all x > 0 NUF(x) = ℵo. That would be wrong even when no gaps 
between the unit fractions existed.
> 
> | Assume otherwise.
> | Assume NUF(x₃) = 3
> |
> | u₁ < u₂ < u₃  are all of
> | the finite unit fractions in (0,x₃)
> |
> | However,
> | ⅟(1+⅟u₁) < u₁ is also
> | a finite unit fraction in (0,x₃)
> | 0 < ⅟(1+⅟u₁) < u₁ < u₂ < u₃ < x₃
> |
> | NUF(x₃) > 3
> | Contradiction.
> 
> Therefore,
> no such x with NUF(x) = 3  exists.

All that is in vain if you accept mathematics, in particular
∀n ∈ ℕ: 1/n - 1/(n+1) > 0.

Regards, WM