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

On 22.12.2024 22:22, Jim Burns wrote:

On 12/22/2024 6:32 AM, WM wrote:

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 infiniteᵂᴹ size or
refuse to describe these transitions at all.

 Alternatively,
set theorists could continue to talk about
what set theorists talk about, instead of
what you (WM) talk about.
 

They do not wish to recognize that their theory is self-contradictory. But perhaps students would be interested. I tell them the following story.
The function E(n) decreases from infinity to zero because in set theory ℕ \ {1, 2, 3, ...} =  { } is an accepted formula.
The set ℕ can get empty by subtracting its elements. Either this is possible one by one, then finite endsegments do exist, or it is only possible be removing (after the first elements one by one) the remaining elements collectively. This shows the existence of numbers which can be handled collectively only.
How should we call them?
Another approach is to call the empty set the limit of the sequence E(n). But note that a limit is a set between which and all terms of the sequence nothing fits. Therefore the limit empty set causes sets with few elements only.
Regards, WM