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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Wed, 21 Aug 2024 20:10:54 -0400
Organization: i2pn2 (i2pn.org)
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On 8/21/24 8:32 AM, WM wrote:
> Le 21/08/2024 à 13:32, Richard Damon a écrit :
>> On 8/21/24 6:44 AM, WM wrote:
>>> Le 20/08/2024 à 23:25, FromTheRafters a écrit :
>>>> WM explained :
>>>>> Le 20/08/2024 à 12:31, FromTheRafters a écrit :
>>>>>> on 8/19/2024, Richard Damon supposed :
>>>>>
>>>>>>> You can not derive a first number > 0 in any of the Number System 
>>>>>>> that we have been talking about, Unit Fractions, Rationals or 
>>>>>>> Reals, so you can't claim it to exist.
>>>>>>
>>>>>> Not in their natural ordering.
>>>>>
>>>>> Dark numbers have no discernible order. It is impossible to find 
>>>>> the smallest unit fraction or the next one or the next one. It is 
>>>>> only possible to prove that NUF(x) grows by 1 at every unit 
>>>>> fraction. It starts from 0.
>>>
>>>> Normally, the unit fractions are listed in the sequence one over 
>>>> one, one over two, one over three etcetera. There is a first but no 
>>>> last. Now you have started from the wrong 'end'
>>>
>>> No, I have started from the other end. It exists at x > 0 because 
>>> NUF(0) = 0.
> 
>> But the other end doesn't "begin" with a first Natural Number Unit 
>> fraction, if it has a beginning that will be a trans-finite number.
> 
> No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all and 
> only reciprocals of natural numbers.
> 
> Regards, WM
> 
> 

Can't be, because if it WAS 1/n, then 1/(n+1) would be before it, and 
thus your claim is wrong. If 1/(n+1) wasn't smaller than 1/n, then we 
just have that 1/n - 1/(n+1) wouldn't be > 0, so it can't be.

BY DEFINITION, there is no "Highest" Natural Number, if n exists, so 
does n+1, and your formula says you accept that n+1 exists, or you 
couldn't use it.

If you don't have that property, you don't have the Natural Numbers.

PERIOD.

DEFINITION.

If you claim your mathematics say it can't be, then your mathematics 
were just proven to not be abble to handle the unbounded set of the 
Natural Numbers.

Sorry, that is just the facts.