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

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 pioneer mathematicians
early on in the development of set theory who were still grasping after
precise notions.

All mathematicians whom you have disqualified above are genuine
mathematicians.

What you perhaps could understand is this simple example: As Cantor said
actual infinity, for instance omegas and alephs and ℕ are fixed
quantities. The set ℕ is invariable.

Maybe Cantor did say this, as a pioneer early on in the development of
set theory.  Things have developed since then, and we see that there is
nothing to be gained by construing infinite sets as "fixed quantities";
there is no mathematical proof where such a concept makes the slightest
difference.

But 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.

Proof:

No, not a mathematical proof.  You have never studied maths to degree
level, and have no idea what a mathematical proof looks like.  Hint:
have a look at some of Jim's posts where he posts rigorous proofs.

{1, 2, 3, ..., 100n} is less than ℕ. That means the set of FISONs will
never cover ℕ, nor will its union reach the invariable quantity..

No, it doesn't mean that at all.  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.

The set of FISONs is only potentially infinite, not a fixed quantity
but growing over all finite bounds.

This "potentially" and "actually" infinite has led you astray, away from
the truth.  They are solely historical notions, with no place in modern
mathematics.  The plain fact is that the set of FISONs is infinite - it
can be put in a 1-1 correspondence with a proper subset of itself.  That
is all.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).