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

On 12/9/2024 4:04 AM, WM wrote:

On 08.12.2024 19:01, Jim Burns wrote:

You (WM) are considering
infinite dark.finite.cardinals,
which do not exist.

>
Then analysis is contradicted in set theory.

Infinite dark.finite.cardinals contradict themselves.
Your two sequences as you have written them
are equal, and have equal limits: the empty set.
⎛ The set of each intersection up.to.an.end.segment
⎜ and
⎜ the set of each end.segment
⎜ both hold
⎜ too.many to have any finite cardinal in common.
⎜
⎜ Only finite.cardinal are in
⎜ the set of finite.cardinals
⎜ and in its end.segments
⎜ and in intersections of them
⎜ and in intersections of intersections of them.
⎜
⎜ The limits
⎜ the sets of common elements
⎜ don't hold any finite.cardinal
⎜ don't hold anything else
⎜ don't hold anything.
⎝ are {} = {}

∀n ∈ ℕ: E(1)∩E(2)∩...∩E(n) = E(n).
The limit of the left-hand side is empty,
the limit of the right-hand side is full,
i.e. not empty.
I do not tolerate that.

I suspect that it is the distinction between
cardinality of limit #⋂{E(i):i} = #{} = 0  and
limit of cardinalities ⋂{#E(i):i} = ⋂{ℵ₀:i} = ℵ₀
which you (WM) are not.tolerating
and which some other posters might have taken you
to be referring to.
There, you do disagree.
And there, you are wrong.