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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Wed, 7 Aug 2024 12:12:29 -0700
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On 8/7/2024 12:05 PM, WM wrote:
> 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.

LOL!!! So, 1.0025 is dark while 1.0024(9) is not?



>>
>> 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
> 
>