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

On 12/28/2024 2:36 PM, Ross Finlayson wrote:

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:

>

[...]

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

>
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,

>
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.
>
Regards, WM
>
Regards, WM
>

>
Just shows that you don't understand *AT ALL* about infinity.
>
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.
>
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.
 

Think of a number that uses a random number per digit for infinity. It still is within the set of unsigned integers. For an interesting rule, the first random digit is not allowed to be zero.
Say, 1875520200019832...
Each digit is random. This is a number and its within the infinite set of unsigned integers.