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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: Tue, 14 Jan 2025 09:02:47 +0100
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Am 13.01.2025 um 21:55 schrieb Chris M. Thomasson:
> On 1/13/2025 9:17 AM, WM wrote:

>> doubling of all natural numbers creates numbers larger than ω.

Idiotic nonsense.

> No. Double all of the natural numbers:
> 
> {1*2, 2*2, 3*2, 4*2, ...} = {2, 4, 6, 8, ...} .
> 
> All of those results are natural numbers. They were already there...

Indeed. (At least in the context of classical mathematics/set theory.)

Using symols:

          An e IN: n*2 e IN .   (*)

Now, by definition (of ω):

          An e IN: n < ω .

Hence (with (*)):

          An e IN: n*2 < ω .

That's why WM's claim "doubling of all natural numbers [results in] 
numbers larger than ω" is idiotic nonsense. For this -as you can see- we 
do not even have to consider "numbers larger than ω".

A simple diagram might be helpful (except for WM):

          1 < 1*2 < 3 < 2*2 < 5 < 3*2 < 7 < ... < ω .

>>> ω is first infiniteᵒʳᵈ.
>>> No infiniteᵒʳᵈ is before ω
>>> No finiteᵒʳᵈ is after ω

Indeed!

>>> distances in ⦅0,ω⦆ multiplied by 2 remain regular distances in ⦅0,ω⦆, not.in ⟦ω,2ω⦆

I guess, this should read: "not.in ⟦ω,ω2⦆".

>> Doubling of all n deletes the odd numbers 

Mückenheim, Du redest wieder mal saudummen Scheißdreck daher. Nichts 
wird "deleted", Du Depp.

Die Menge G = {n*2 e IN : n e IN} ist einfach eine (echte) Teilmenge der 
natürlichen Zahlen, die (per definitionem) genau die sog. /geraden 
(natürlichen) Zahlen/ enthält.

>> but cannot change the number of numbers,

Das wiederum ist richtig. Es gilt nämlich (wegen IN ~ G):

              |IN| = |G|
bzw.
              card(IN) = card(G) .

>> therefore creates even numbers.

Nö, die werden nicht "created", Du Depp, sondern lediglich mit Hilfe des 
sog. Aussonderungsaxioms aus der Menge der natürlichen Zahlen, IN, 
"ausgesondert": D. h. die waren vorher schon da und in IN enthalten 
(Hint: G c IN.)

Das "Aussonderungsaxiom" sichert die Existenz der Menge G = {n*2 e IN : 
n e IN}.

>> They do not fit below ω 

Doch das tun sie, weil sie als natürliche Zahlen natürlich (sic!) 
kleiner als ω sind.

Hint:         1 < 1*2 < 3 < 2*2 < 5 < 3*2 < 7 < ... < ω .

>> Remember: [no ordinal number] fits between them and ω.

In der Tat: ~Eo e ORD: An e IN: n*2 < o < ω .

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