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

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"