Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/30/2024 2:07 PM, WM wrote:

On 30.11.2024 18:32, Jim Burns wrote:

Apparently, what you (WM) call "the intersection"
is each of infinitely.many intersections,

>
and their limit.

some infinite, some empty,
"changing" from one to another,
in a manner you accept or you do not accept.

The limit of
all the end segments of the finite cardinals
is one of those intersections.
It is the set of common finite.cardinals,
  common to each end.segment, that is, their intersection.
It is empty.
Intersections of some other end.segment.collections
are infinite. And still others also empty.

Our sets do not change.

>
But there is a sequence of endsegments
E(1), E(2), E(3), ...

With an empty set of common finite.cardinals.

and a sequence of their intersections
E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ... .

With an empty set of common finite.cardinals.

Both are identical -
from the first endsegment
on until every existing endsegment.

Yes, they are identical. And identically empty.
----
Consider a finite.cardinal ξ
There are only finitely.many
  fore.segments ⟦0,β⟧ ⊆ ⟦0,ξ-1⟧ which ξ is not.in.
There only finitely.many
  end.segments E(β+1) = ℕ\⟦0,β⟧ which ξ is in.
Consider a collection Ends of end.segments.
If Ends is an infinite collection,
then
⎛ there are more end.segments in Ends
⎜  than there are end.segments holding ξ
⎜ ξ is not common to all end.segments in Ends
⎝ ξ is not.in ⋂Ends
The same argument applies to each finite cardinal.
If Ends is an infinite collection,
then
⎛ Each finite.cardinal is not.in ⋂Ends
⎝ ⋂Ends = {}
Each infinite collection of
  end.segments of
   finite.cardinals
has an empty intersection.