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

On 18.12.2024 02:16, Jim Burns wrote:

On 12/17/2024 5:12 PM, WM wrote:

We have the sequence of
intersections of endsegments
f(k) = ∩{E(1), E(2), ..., E(k)}
with E(1) = ℕ
and the definition of that function
∀k ∈ ℕ :
∩{E(1), E(2), ..., E(k+1)} =
∩{E(1), E(2), ..., E(k)} \ {k}
and the fact that
∩{E(1), E(2), ...} is empty.
More is not required to prove
the existence of finite endsegments.

 Here is a counter.example to your claimed requirement:
There are no finite.cardinals common to
all the infinite end.segments of finite.cardinals.

There are no definable cardinals common to all endsegments of definable cardinals. Finite cardinals are in all non-empty endsegments.

The infinite end.segments of finite.cardinals
do not include any finite end.segments and
they have an empty intersection.

Explicitly wrong. As long as only infinite endsegments are concerned their intersection is infinite.

----
More generally,
for limit.set Lim.⟨A₀,A₁,A₂,…⟩ of

There are no limits involved. Cantor uses "all natural numbers" (no limit) in his bijections, that means to use them all coming out of endsegments. None remains. The intersection of all endegments is empty. But it gets empty one by one.
Regards, WM