Replacement of Cardinality

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?
Not all infinite sets can be compared by size, but we can establish some useful rules
_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.
_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. _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.
Regards, WM