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Subject: Re: Replacement of Cardinality
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Date: Sat, 27 Jul 24 11:09:40 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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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