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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Mon, 2 Dec 2024 06:50:37 +0100
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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 .