Re: how

On 4/23/24 3:34 PM, WM wrote:

Le 23/04/2024 à 01:01, Richard Damon a écrit :

On 4/22/24 10:15 AM, WM wrote:

 

The results cannot be compressed to the interval (0, ω) of the set { 1, 2, 3, ...}. This shows that new numbers are generated by multiplication.
>

Of course they can be compressed into the interval (0, ω), as every finite number n < ω, when doubled results in a finite number 2n which is also < ω.

 Try to map the closed interval [0, ω]*2 = [0, ω*2].
If [0, ω) --> [0, ω) and ω*2 --> ω*2, then ω*2 is the only image point in (ω, ω*2]. Infinitely many points remain empty. Crippled mathematics. Ugly. Inacceptable.
 Regards, WM
  

Why?
[0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural numbers (what [0, ω) represents) are closed under multiplication.
The fact that a mixed set of two different classes of ordinals ends up with two different classes of ordinals isn't surprizing.
The fact that there are "gaps" in the result isn't surprising, as we see the same gaps in the finite part of the set:
0, 1, 2, 3 ... *2 => 0, 2, 4, 6, ...
so the fact that 1 disappeared into a gap in the result says the gap between the finites and ω*2 is reasonable.