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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Sat, 23 Nov 2024 06:32:09 +0100
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Am 23.11.2024 um 06:18 schrieb Moebius:
>>> On 22.11.2024 16:11, joes wrote:
>>>> Am Fri, 22 Nov 2024 15:51:11 +0100 schrieb WM:
> 
>>>>> the sets of naturals and of prime numbers [can] cover each other.
>>>>>
>>>> As it should. You can give each prime an index.
> 
> Indeed! The two formulas
> 
> | p(1) = min P
> | p(n+1) = min {p e P : p > p(n)}   (for all n e IN)
> 
> (recursively) define the function p: IN --> P. Where IN is the set of 
> all natural numbers and P is the set of all prime numbers.
> 
> Hint: p(1) = 2, p(2) = 3, p(3) = 5, ...
> 
> Actually, if p e P, then there is an (index) n e IN such that p(n) = p.
> 
> (It's easy to prove that p: IN --> P is a bijection.)

p(n) is the n-th prime number in the sequence of prime numbers ordered 
by size.

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