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NNTP-Posting-Date: Mon, 30 Dec 2024 01:15:48 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary, effectively)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 29 Dec 2024 17:15:31 -0800
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On 12/29/2024 01:41 PM, Jim Burns wrote:
> On 12/29/2024 3:54 PM, Ross Finlayson wrote:
>> On 12/29/2024 12:34 PM, Jim Burns wrote:
>>> On 12/28/2024 10:54 PM, Ross Finlayson wrote:
>>>> On 12/28/2024 04:22 PM, Jim Burns wrote:
>>>>> On 12/28/2024 5:36 PM, Ross Finlayson wrote:
>
>>>>>> Then it's like
>>>>>> "no, it's distribution is non-standard,
>>>>>> not-a-real-function,
>>>>>> with real-analytical-character".
>>>>>
>>>>> Which is to say,
>>>>> "no, it isn't what it's described to be"
>>>>
>>>> You already accept
>>>
>>> No.
>>> You (RF) are greatly mistaken about
>>> my (JB's) position with regard to
>>> infinitely.many equal real.number steps
>>> from 0 to 1
>>>
>>> My position is and has been that they don't exist.
>>>
>>>> You already accept that the "natural/unit
>>>> equivalency function" has range with
>>>> _constant monotone strictly increasing_
>>>> has _constant_ differences, _constant_,
>>>> that as a cumulative function, for a
>>>> distribution, has that relating to
>>>> the naturals, as uniform.
>>>
>>> My position, expressed in different ways,
>>> is and has been that,
>>> for each positive real x,
>>> a finite integer n exists such that
>>> n⋅x > 1
>>>
>>> That conflicts with the existence of
>>> infinitely.many equal real.number steps
>>> from 0 to 1
>>
>> Oh, you had that [0/d, 1/d, 2/d, ... oo/d]
>> was a thing,
>
> No.
> What I had was that
> ⋃ᵢ₌᳹₀[0/i,1/i,2/i,...,i/i]
> is a thing,
> but the thing which it is
> is not an interval of real numbers.
>
> That's still my position.
>
>> So, yeah, it conflicts with yourself,
>> yet, that's what you said.
>
> No,
> it isn't.
>
>

It's also an equi-partitioning.

Here of course it IS a continuous domain
or model of real numbers, ran(EF),
because being a distribution of the
natural integers at uniform random.


Well, go ahead then and come up with
some reason why it's not an equi-partitioning
as it fulfills otherwise being a CDF.

Figuring you just can't accept it, ....


It's like the not.first.false <-> not.ultimately.untrue
bit and the "yin-yang ad-infinitum" bit, showing that
it's not.ultimately.untrue that the inductive account
is not accurate.


Here though the "extent, density, completeness, measure"
or "extent, density, gaplessness, measure", are considered
making for a continuous domain, yet at least the larger
machine reasoning inference systems agree that's so.