Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/22/2024 12:47 PM, Ross Finlayson wrote:

On 11/22/2024 12:37 PM, Chris M. Thomasson wrote:

On 11/22/2024 6:51 AM, WM wrote:

On 22.11.2024 13:32, joes wrote:
 > Am Fri, 22 Nov 2024 13:00:52 +0100 schrieb WM:
>
 >>>>>>>> The number of ℕ \ {1} is 1 less than ℕ.
 >>>>>>> And what, pray tell, is Aleph_0 - 1 ?
 >>>>>> It is "infinitely many" like Aleph_0.
 >>> Thanks for agreeing with |N| = |N\{0}|.
 >> Of course. ℵo means nothing but infinitely many.
 > Good. Then we can consider those sets to have the same number.
 >
That is the big mistake. It makes you think that the sets of naturals
and of prime numbers could cover each other.

>
prime numbers are a sub set of the naturals. They are both infinite.

 Finite, though large, "sets", as though all the relations
among them besides just "the set of", make it so in the
asymptotics, that it's possible to work up when the
density of primes, which is kind of known and on the
order of log n, vis-a-vis pi^2/6 and co-primes, have
it so that in some "practically" or "effectively"
"un-bounded", if that's a short-hand interchangeable
with "infinite", have only finitely many primes.
 Maybe one at infinity, ....

[...]
How can there be one prime at infinity? That's like saying there is a natural number at infinity. There is no largest natural just that there is no largest prime. So, if you artificially say this prime is at infinity you just went into finite mode!