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Subject: Re: Replacement of Cardinality
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Date: Wed, 07 Aug 24 19:05:05 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 07/08/2024 à 20:49, Jim Burns a écrit :
> On 8/7/2024 9:05 AM, WM wrote:
>> Le 07/08/2024 à 04:36, Richard Damon a écrit :
>>> On 8/6/24 4:35 AM, WM wrote:
> 
>>>> Right. But with NUF(x) = 1 ==> INVNUF(1) = x we get
>>>> ∃u ∈ ⅟ℕ, u < x, ∀y > x = INVNUF(1).
>>>
>>> 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.
>> 
>> Peano is not valid for all dark numbers.
> 
> For each real x > 0
> there are ℵ₀.many visibleᵂᴹ unit.fractions
> between x and 0

For each visible real.
> 
> There is no real x > 0 such that
> there are fewer than ℵ₀.many visibleᵂᴹ unit fractions
> between x and 0

They exist but are dark.
> 
> Darkᵂᴹ numbers do not make _fewer_ visibleᵂᴹ numbers,
> do they?

No.
> 
> If darkᵂᴹ numbers are what's between [0,1] and (0,1]

No that is not the case.

Regards, WM