Re: Replacement of Cardinality

Le 26/07/2024 à 23:57, joes a écrit :

Am Fri, 26 Jul 2024 16:31:19 +0000 schrieb WM:

It is strange that blatantly false results as the equinumerosity of
prime numbers and algebraic numbers could capture mathematics and stay
there for over a century. But by what meaningful mathematics can we
replace Cantor's wrong bijection rules?

Juste because it doesn't match your intuition doesn't mean it's not
useful.

Cantor has been disproved in different ways. See for instance https://www.academia.edu/91188101/Proof_of_the_existence_of_dark_numbers_bilingual_version_

 

Not all infinite sets can be compared by size, but we can establish some
useful rules

that you would like instead.

There are more natural numbers than prime numbers. That is fact.

_The rule of subset_ proves that every proper subset has less elements
than its superset. So there are more natural numbers than prime numbers,
|ℕ| > |P|, and more complex numbers than real numbers.  Even finitely
many exceptions from the subset-relation are admitted for infinite
subsets. Therefore there are more odd numbers than prime numbers.

What exceptions do you mean?

The exception prime number 2 is not an odd number.

This immediately creates as many sizes as there are naturals, one for
each of your endsegments.

_The rule of symmetry_ yields precisely the same number of reals in
every interval (n, n+1] and with at most a small error same number of
odd numbers and of even numbers in every finite interval and in the
whole real line.

How small an error?

Only 1 or 2 depending on the chosen interval. In the interval (0, 3] there are two odd natnumbers but only one even natnumber.
Regards, WM