Re: how

Le 24/04/2024 à 22:07, "Chris M. Thomasson" a écrit :

On 4/24/2024 10:16 AM, WM wrote:

Le 24/04/2024 à 04:01, Richard Damon a écrit :

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.

 

Why?

 Continuity.

>
[0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural numbers (what [0, ω) represents) are closed under multiplication.

 Impossible if (0, ω) is completely filled. Every 2n is larger than n.

[...]
 How can it be 100% completely filled when its unbounded?

Ask Cantor or Bolzano. It can also be completely counted - according to Cantor.
But don't ask modern set theorists. They are either too stupid or too dishonest to give the correct answer.
Regards, WM