Re: More complex numbers than reals?

Am 11.07.2024 um 21:39 schrieb Chris M. Thomasson:

Are the gaps [between] prime numbers "random" wrt their various length's?

In a certain sense (maybe) "yes", but there are "rules". :-P
For example, for each and ever natural number n > 1 there's a prime between n and 2n. :-)
See: https://en.wikipedia.org/wiki/Bertrand%27s_postulate
Moreover: "The prime number theorem, proven in 1896, says that the average length of the gap between a prime p and the next prime will asymptotically approach ln(p), the natural logarithm of p, for sufficiently large primes. The actual length of the gap might be much more or less than this. However, one can deduce from the prime number theorem an upper bound on the length of prime gaps[.]"
Source: https://en.wikipedia.org/wiki/Prime_gap

(2, 3) has no gap wrt the naturals, however, (3, 5) does [...].

For all n e IN, n > 1, there is a prime between n and 2n.
Let's check it for some n:
n = 2: 2, *3*, 4
n = 3: 3, 4, *5*, 6
n = 4: 4, *5*, 6, *7*, 8
n = 5: 5, 6, *7*, 8, 9, 10
:
On the other hand, it's NOT known if for all natural number n there is a prime between n^2 and (n+1)^2.
See: https://en.wikipedia.org/wiki/Legendre%27s_conjecture
Fascinating, isn't it?
_______________
Back to countably infinite sets. :-)

https://en.wikipedia.org/wiki/Countable_set

 We can index the primes:
 [0] = 2
[1] = 3
[2] = 5
[3] = 7
...

Exactly.
This means (using math terminology) that there is a bijektion between IN and P (the set of primes).
Hence the set of primes is countably infinite.