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

On 1/5/25 12:52 PM, WM wrote:

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.

And thus, the SET of Natural Numbers doesn't "grow" as you discover more Natural Numbers, tbey were there before, but you hadn't "seen" them

>

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.

Which doesn't prove your point.

 

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, there is a LAW that shows that the union of ALL the FISONs will reach what you say it doesn't.
The fact that your wording even admits that you logic just quits before you get to the end of the task proves you stupidity.

 

No, not a mathematical proof.

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

But your "theorem" is a LIE, you can't PROVE it, because you don't actually know what you are talking about.
Your "theorem" is just your own misconceptions about the nature of numbers, based on just the smallness of your mind.

{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.

No, your "theorem" is just a lie that comes out of your own stupidity.
That you call it a "theorem" just PROVES that you don't know what you are talking about, or how mathematics works.
You are just proving you are too stupid to understand that you are too stupid to understand.

 Regards, WM