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

Am Sat, 28 Dec 2024 17:50:49 +0100 schrieb WM:

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;

 
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.

A relative size is useless because it is still exactly as infinite,
i.e. countable. You seem to think there are only finitely many naturals
and therefore FISes. Of course, if you only consider fin. many, you
don’t have all.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.