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

On 22.11.2024 22:49, FromTheRafters wrote:

After serious thinking WM wrote :

On 22.11.2024 16:11, joes wrote:

Am Fri, 22 Nov 2024 15:51:11 +0100 schrieb WM:

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.

As it should. You can give each prime an index.

>
But not every index a prime.

 Why not? It looks like a bijection to me.

>
Up to every prime, yes. But not for all numbers.
>
Let every unit interval after a natural number on the real axis be coloured white with exception of the intervals after the prime numbers which are coloured red. It is impossible to shift the red intervals so that the whole real axis becomes red. Every interval (10n, 10 (n+1)] is deficient - on the whole real axis.