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

On 05.01.2025 18:35, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

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";

It is a precondition of set theory.

there is no mathematical proof where such a concept makes the slightest
difference.

Every proof in set theory is based on the invariability of sets.

 

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.

It is not meaningless but an abbreviation for the fact that multiplication by 100 does not reach or surpass ℕ.

Finally, it is
wrong, absurdly wrong.  The union of all FISONs _is_ N.

That is a a dogma of matheologioy disproved by the fact that every union of FISONs which stay below a certain threshold stays below that threshold. Every FISON stays below 1 % of ℕ.

No, not a mathematical proof.

You mathematical "proofs" contradict the simple theorem stated above. Therefore they are invalid.

{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.

My theorem is better understood by school students not yet stultified by set theory.
Regards, WM