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

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, ....
It's similar the fundamental theorems of arithmetic,
like this, and algebra, this way, when there's too
many to count, yet what makes factors or roots of it,
has a finite amount above the, "floor", as it were.
This is why there's "potential and actual" and
"practical and effective" "in-finities", with
regards to the expectations and guarantees of
what's finite and what's in-finite, with regards
to concrete mathematics and asymptotics, and
the independence of number theory with regards
to fragments and extensions of some soi-disant
"standard, integers.
Laws of large numbers and concrete implicits, ....
Of course there are quite usual proofs of there
being infinitely-many primes, and more and less
naive ones, as with regards to those assumptions
the more or less naive, going into the "structure"
the thing, often with regards to "products of all
primes, +- 1", vis-a-vis, "prime or not".
Anyways though these are reasons why it's both,
not reasons why it's wrong.