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

On 12/24/24 5:53 AM, WM wrote:

On 23.12.2024 19:20, Jim Burns wrote:

On 12/23/2024 6:18 AM, WM wrote:

 

The empty limit set is caused by
each finiteⁿᵒᵗᐧᵂᴹ.cardinal being not.in
almost all of the end.segments
(in only finitely.many end.segment.exceptions)

>
And why is this so?
Because
every endsegment removes exactly one number from the content.
There is a bijection between
endsegments and removed numbers.

>
There is a bijection between
more.than any.finiteⁿᵒᵗᐧᵂᴹ.cardinal.many
end.segments and
more.than any.finiteⁿᵒᵗᐧᵂᴹ.cardinal.many
removed numbers?

 Nonsense.
 

Have you told your students yet?

 Of course not. There is a bijection {n} <--> E(n). No cardinal number ℵ₀ is involved.
 Regards, WM
 

Except that both sets, this size of the Natural Numbers and the size of the set of E(n), are Aleph_0.
Once you mention "size" or "number of elements" you reference that value indirectly.