Re: how

On 4/24/24 1:16 PM, 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.

Who says we have that sort of "Continuity"?
Infinite numbers don't behave the same as what we expect out of finite numbers.

>
[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.

Right, 2n is greater than n but still finite and < ω. That is the nature of unbounded sets, and blows up your finite logic.

>
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, ...

 Here are always gaps of one unit. Between ω and ω2 is an infinite emptiness.

Right, because they are a different type of Ordinals. Between ω and ω+1 is one step, between 0, ω, 2ω, 3ω, ... is a first order infinite step.

 Crippled ugly mathematics.

Just beyound your ability to comprehend. You just can't appreciate the beauty of infinity because it blows out your mind that is just too small.

 Regards, WM