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NNTP-Posting-Date: Sat, 28 Dec 2024 22:36:19 +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 14:36:23 -0800
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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.