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

On 28.12.2024 18:48, joes wrote:

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

 

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.

A relative size is very useful in order to correct the fantasy claim
that the union of all FISONs covers the infinite gap between all FISONs
and ω.