Re: because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

Le 23/04/2024 à 05:51, Jim Burns a écrit :

On 4/22/2024 1:56 PM, WM wrote:

Le 22/04/2024 à 19:32, Jim Burns a écrit :

On 4/22/2024 11:35 AM, WM wrote:

 

0, 1, 2, 3, ...,   w  = X
|  |  |  |  |||    |
0, 2, 4, 6, ..., w*2  = Y

>
The original set fits in (0, ω).
The doubled set doesn't.

>
⦅0,ω+ω⦆ fits ⦅0,ω⦆

>
No. ω+ω = ω*2.

 Yes. ω+ω = ω⋅2
⦅0,ω+ω⦆ fits ⦅0,ω⦆
⦅0,ω⋅2⦆ fits ⦅0,ω⦆

Use closed intervals: [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 the interval (ω, ω*2]. Infinitely many points remain empty. Ugly mathematics.
 

ω+i ⟼ 2⋅i

ω+i is not mapped.
Regards, WM