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

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

On 12/28/24 11:50 AM, WM wrote:

On 28.12.2024 15:12, Jim Burns wrote:

On 12/27/2024 5:24 PM, Ross Finlayson wrote:

On 12/27/2024 01:00 PM, Jim Burns wrote:

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[...]

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The, "almost all", or, "almost everywhere",
does _not_ equate to "all" or "everywhere",

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Correct.
⎛ In mathematics, the term "almost all" means
⎜ "all but a negligible quantity".
⎜ More precisely, if X is a set,
⎜ "almost all elements of X" means
⎜ "all elements of X but those in
⎜ a negligible subset of X".
⎜ The meaning of "negligible" depends on
⎜ the mathematical context; for instance,

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A good example is the set of FISONs. Every FISON contains only a
negligible quantity of natural numbers. A generous estimation is:
Every FISON contains less than 1 % of all natural numbers. There is no
FISON that contains more than 1 %. Therefore the union of all FISONs
contains less than 1 % of all natural numbers. Outside of the union of
FISONs are almost all natural numbers.
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Regards, WM
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Regards, WM
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Just shows that you don't understand *AT ALL* about infinity.
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Every Natural Number is less that almost all other natural numbers, so
its %-tile of progress is effectively 0, but together they make up the
whole infinite set.
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The fact that you mind can't comprehend that just proves your stupidity.

Consider a random uniform distribution of
natural integers, same probability of each integer.
Now, you might aver "that can't exist, because it
would be non-standard or not-a-real-function".
Then it's like "no, it's distribution is non-standard,
not-a-real-function, with real-analytical-character".
So, beyond the idea of small numbers that grow as
"the law of large numbers", is that there are others,
and furthermore, what in probability theory is a
remarkable counterexample to the uniqueness of
probability distributions, has that the plain
old "natural/unit equivalency function" is among
distributions of the naturals at uniform random.
Trust me, you'll run out of fingers trying to count that.