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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, 6 Jan 2025 05:17:25 +0100
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Am 05.01.2025 um 18:35 schrieb Alan Mackenzie:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> On 05.01.2025 12:28, Alan Mackenzie wrote:
> 
>>> The only people who talk about "potential" and "actual" infinity are
>>> non-mathematicians who lack understanding, and [...]

>> All mathematicians whom you have disqualified above are genuine
>> mathematicians.

Yes, but you are NOT, Mückenheim, and it shows!

Even worse, you are a mathematical crank.

Example:

>> [...] all finite initial segments of natural numbers FISONs {1, 2, 3,
>> ..., n} cover less than 1 % of ℕ.
> 
> That is a thoroughly unmathematical statement.  To talk about 1% of an
> infinite set is meaningless.  To say "cover" in the context of set
> theory rather than topological spaces is inappropriate.  Above all, to
> say "all finite initial segments" is unmathematical, since what is meant
> is not the set of FISONs, but the union of FISONs.  Finally, it is
> wrong, absurdly wrong.  The union of all FISONs _is_ N.

Indeed.

>> Proof:

As if.

> No, not a mathematical proof.  You have never studied maths to degree
> level, and have no idea what a mathematical proof looks like.  [...]

Right.

> [...] The set of FISONs does indeed "cover"
> N, in the sense that their union is equal to N.  A proof of this is
> trivial, well within the understanding of a school student studying
> maths.

Satz: U{A(k) : k e IN} = IN.

Beweis: Für alle n e IN ist n e A(n+1). D. h. für alle n e IN gibt es 
ein k e IN mit n e A(k). Also gilt für alle n e IN: n e U{A(k) : k e 
IN}. D. h. IN c U{A(k) : k e IN}. Da aber (wegen An e IN: A(n) c IN) 
auch U{A(k) : k e IN} c IN ist, gilt U{A(k) : k e IN} = IN. qed

>> The set of FISONs is only potentially infinite, not <bla>

There are no "potentially infinite" sets. Actually, only finite and 
infinite sets (in the context of set theory).

> This "potentially" and "actually" infinite has led you astray, away from
> the truth.  They are solely historical notions, with no place in modern
> [classical] mathematics [i. e. set theory + classical logic --moebius].
> The plain fact is that the set of FISONs is infinite [...]

Indeed!

Satz: "Die Menge aller FISONs ist abzählbar unendlich."

Beweis:

Es gibt eine Bijektion zwischen IN und der Menge aller FISONs {A(n) : n 
e IN} [mit A(n) := {m e IN : m < n} (n e IN)]. Nämlich die Abbildung f: 
IN --> {A(n) : n e IN}, die durch f(n) = A(n) für alle n e IN, definiert 
ist. f ist trivialerweise surjektiv. Die Injektivität von f ergibt sich 
aus f(n1) = A(n1) = {m e IN : m < n1} c_echt {m e IN : m < n2} = A(n2) = 
f(2) für n1,n2 e IN mit n1 < n2. Denn daraus folgt f(n1) =/= f(2) für n1 
=/= n2.

Wir haben also IN ~ {A(n) : n e IN} gezeigt.

Daraus folgt card(IN) = card({A(n) : n e IN}) und mit card(IN) = aleph_0 
schließlich card({A(n) : n e IN}) = aleph_0. qed

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