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

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
.
.
.