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

On 12/28/2024 07:54 PM, Ross Finlayson wrote:

On 12/28/2024 04:22 PM, Jim Burns wrote:

On 12/28/2024 5:36 PM, Ross Finlayson wrote:

On 12/28/2024 11:17 AM, Richard Damon wrote:

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[...]

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Consider
a random uniform distribution of natural integers,
same probability of each integer.

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A probability.measure maps events
(such as: the selection of an integer from set S)
to numbers in real.interval [0,1]
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For each x ∈ (0,1]
there is a finite integer nₓ: 0 < ⅟nₓ < x
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⎛ 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.
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Therefore,
there is no uniform probability.measure on
the non.negative integers.
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Now, you might aver
"that can't exist, because it would be
non-standard or not-a-real-function".

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I would prefer to say
"it isn't what it's describe to be,
because what's described is self.contradictory".
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Then it's like
"no, it's distribution is non-standard,
not-a-real-function,
with real-analytical-character".

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Which is to say,
"no, it isn't what it's described to be"
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>

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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.
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And that they always add up to 1, ....
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That most certainly is among the definitions
of a distribution, the probabilities even
range between 0 and 1.
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So, even though you refuse that this is
a real function, because it's not, yet
it's also a distribution, which it is.
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"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.
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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".
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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.
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So that ZF isn't internally inconsistent
with respect to real numbers, ....
... and continuous domains.
Of course one could always just write it
in some otherwise equi-interpretable
"theory of one relation" yet it's so
that ZF and ZFC are the most well-known
and well-explored axiomatic set theories.
"Pick either: get both."