Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, effectively)

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.