Re: how

Le 18/04/2024 à 00:59, Richard Damon a écrit :

On 4/17/24 2:49 PM, WM wrote:

The size of ℕ is |ω|. That means ℕ extends on the ordinal line from 0 to ω. By multiplying every natural number the extendion is doubled. That is mathematics. Every contrary opinion is foolish.

 Nope, the size of ℕ, that is |ℕ| is aleph_0. ω is an ORDINAL number

on the ordinal axis beyond all natural numbers with nothing else before it.
ℵo = |ℕ| = |ω|.
 

(showing order), not a cardinal number (showing size). This means that the first transfinite ordinal above the set ℕ would be the value ω. So yes, as you say below, there are no "finite" numbers between the set ℕ and the value ω, but since ℕ has no "highest" member there is no "predecessor" to ω, just as there is no predecessor to 0 in the Natural Numbers.

Nonsense.
ω follows upon all natural numbers. There is nothing between them and ω.
Regards, WM