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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 06:18:35 +0100
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>> 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.)