Re: how

On 4/23/24 2:42 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.
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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 points many remain empty. Crippled mathematics. Ugly. Inacceptable.

But true. Since the interval [0, ω] contains two different classes of values (all the finite ordianl natural numbers, and then one transfinite ordinal value) it makes sense that the result will have two classes of values, and they may behave differently.
so:
[0, ω]*2 = { [0, ω), ω }*2 -> { [0, w), 2ω }
using the fact that the Natural Numbers are closed under multiplication.

 Regards, WM
   

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Name one that can't be!
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Your logic just blows you up because it doesn't understand the nature of unbounded infinite sets.
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You invent "dark" numbers, because you can't look at the nature of the unbounded because you are blinded by your finite logic that has blown your logic to smiterines because you used it when it can't work.