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

Am 02.12.2024 um 00:11 schrieb Chris M. Thomasson:

On 11/30/2024 3:12 AM, WM wrote:

Finite initial segment[s]: F(n) = {1, 2, 3, ..., n}    (n e IN).

Endsegment[s]: E(n) = {n, n+1, n+2, ...}    (n e IN).

Right ... where
{1, 2, 3, ..., n} := {m e IN : m <= n}}
and
{n, n+1, n+2, ...} := {m e IN : m >= n}} .

The sequence of endsegments decreases. (WM)

>
In what manner [is it] decreasing?

          An e IN: E(n+1) c E(n) .

The [endegmanets] are losing elements, one after the other:
∀k ∈ ℕ : E(k+1) = E(k) \ {k}

Indeed.

But each endsegment has only one element less than its predecessor.

If you say so. Still for all n e IN: card(E(n)) = aleph_0.
Himt: Infnitely many <whatevers> "minus" one <whatever> are still infinitely many <whatevers>.

When you filter out the FISON, the rest, the tail, as a set, stays the same size of aleph_zero.

Right.
An e IN: |IN \ F(n)| = aleph_0 .