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NNTP-Posting-Date: Sun, 29 Dec 2024 03:54:50 +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: Sat, 28 Dec 2024 19:54:49 -0800
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On 12/28/2024 04:22 PM, Jim Burns wrote:
> On 12/28/2024 5:36 PM, Ross Finlayson wrote:
>> On 12/28/2024 11:17 AM, Richard Damon wrote:
>
>>> [...]
>>
>> Consider
>> a random uniform distribution of natural integers,
>> same probability of each integer.
>
> A probability.measure maps events
> (such as: the selection of an integer from set S)
> to numbers in real.interval [0,1]
>
> For each x ∈ (0,1]
> there is a finite integer nₓ: 0 < ⅟nₓ < x
>
> ⎛ Assume a uniform probability.measure
> ⎜ P: 𝒫(ℕ) -> [0,1]
> ⎜ on ℕ the non.negative integers.
> ⎜ ∀j,k ∈ ℕ: P{j} = P{k} = x
> ⎜ x ∈ [0,1]
> ⎜
> ⎜ If x = 0, Pℕ = ∑ᵢ₌᳹₀P{i} = 0 ≠ 1
> ⎜ P isn't a probability.measure.
> ⎜
> ⎜ If x > 0,  0 < ⅟nₓ < x
> ⎜ P{0,…,nₓ+1} > (nₓ+1)/nₓ > 1
> ⎝ P isn't a probability.measure.
>
> Therefore,
> there is no uniform probability.measure on
> the non.negative integers.
>
>> Now, you might aver
>> "that can't exist, because it would be
>> non-standard or not-a-real-function".
>
> I would prefer to say
> "it isn't what it's describe to be,
> because what's described is self.contradictory".
>
>> 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 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.

And that they always add up to 1, ....


That most certainly is among the definitions
of a distribution, the probabilities even
range between 0 and 1.

So, even though you refuse that this is
a real function, because it's not, yet
it's also a distribution, which it is.

"Standardly as a limit of functions"
if you won't, like Dirac's unit impulse
function, never actually so esxcept its completion,
yet necessarily actually so in the derivations
that depend on it, like Fourier-style analysis,
"real analytical character", say.

Then also that it really is "a continuous
domain" and "a discrete distribution",
just keeps pointing out how special it is,
"the natural/unit equivalency function".


One of at least three set-theoretic accounted
models of continuous domains, and establishing
that there are non-Cartesian functions via
an anti-anti-diagonal-argument.