Re: Replacement of Cardinality

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.

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

that you would like instead.

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

_The rule of construction_ yields the numbers of integers |Z| = 2|ℕ| + 1
and of fractions |Q| = 2|ℕ|^2 + 1 (there are less rational numbers).
Since all products of rational numbers with an irrational number are
irrational, there are many more irrational numbers than rational
numbers.

Only this leads to some contradictions depending on the construction.

_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?

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.