Re: Replacement of Cardinality

On 7/26/24 12:31 PM, WM wrote:

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

The problem is that there can be infinite sets constructed different ways that give different answers.
By your logic, if you take a set and replace every element with a number that is twice that value, it would by the rule of construction say they must be the same size.
But that resultant set is the evens, which can also be shown by your logic to have less elements than the Natural Numbers they were made from by doubling, so a set is smaller (or larger) than itself.
This is the sort of error that comes when you try to use logic designed for "finite" sets with infinite sets.
This shows that infinite sets just must have some different rules.