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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Replacement of Cardinality
Date: Sat, 17 Aug 2024 12:21:02 -0400
Organization: i2pn2 (i2pn.org)
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On 8/17/24 11:56 AM, FromTheRafters wrote:
> Richard Damon formulated on Saturday :
>> On 8/17/24 9:30 AM, WM wrote:
>>> Le 16/08/2024 à 19:54, Jim Burns a écrit :
>>>> On 8/16/2024 1:00 PM, WM wrote:
>>>
>>>>> Start from x = 0.
>>>>> The increase cannot be more than 1.
>>>>> ∀n ∈ ℕ: 1/n - 1/(n+1) > 0
>>>>
>>>> Start from x = 0
>>>> The increase cannot be less than 2
>>>
>>> That violates ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 .
>>>
>>>> ∀ᴿx > 0:  x  >  ⅟nₓ  >  ⅟(nₓ+1)
>>>> nₓ = ⌊1+⅟x⌋
>>>
>>> That holds only for definable numbers.
>>>
>>> Regards, WM
>>>
>>>
>>
>> Which is all of the Rational Numbers, and thus all of the unit fractions.
>>
>> Thats your problem, you don't seem to know how to definie numbers.
> 
> Worse than that, he tries to make a set of undefined/undefinable numbers 
> which he calls dark numbers while sets have well defined elements.

No, he can have his "dark numbers", but either they are the result of 
his system just not being able to handle unbounded sets, or they are the 
beyond-finite numbers that exist in the space between the Natural 
Numbers and simple infinities, or between 0 and the positive finite numbers.

There are number system that define these sorts of numbers, and the 
basic logic that handles the unbounded finites can't deal with them, so 
they could be considered "dark" and undefinable.

Of course, since his logic can't even handle the unbounded numbers, it 
can't handle them at all.