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Subject: Replacement of Cardinality
Newsgroups: sci.logic
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Date: Fri, 26 Jul 24 16:31:19 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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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