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

On 22.12.2024 11:16, Jim Burns wrote:

What does 'finite' mean?

A set is finite if it contains a visible natural number of elements.
All natural numbers are finite but the realm of dark numbers appears like infinity.

E(n+2) is the set of all finite.cardinals > n+2
E(n+1)  =  E(n+2)∪{n+2}
 E(n+2)∪{n+2} isn't larger.than E(n+2)

Wrong.

#E(n+2) isn't any of the finite.cardinals in ℕ

It is an infinite number but even infinite numbers differ like |ℕ| =/= |ℕ| + 1. You must not defend Cantor's nonsense by Cantor's nonsense. All "countable cardinalities" are the same because only potential infinity is utilized.

 E(n+2) isn't any of the sets
smaller.than fuller.by.one sets.
 E(n+2) is emptier.by.one than E(n+1), but
E(n+2) isn't smaller.by.one than E(n+1).

Wrong.
The function E(n) decreases from infinity to zero by single steps of height 1 like the function NUF(x) increases by single steps of height 1. Set theorists must accept magic steps of infinite size or refuse to describe these transitions at all.
Regards, WM