Re: how

Le 12/04/2024 à 20:57, Jim Burns a écrit :

On 4/12/2024 9:18 AM, WM wrote:

usedᵂᴹ or not.usedᵂᴹ
not.exists

All individually usable numbers satisfy
∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
All collectively useable numbers satisfy
|ℕ \ {1, 2, 3, ...}| = 0

What elements fall between ω and ω*2?

 ω+1 ω+2 ...

What elements of {1, 2, 3, ..., ω}*2 fall between ω and ω*2?
Their distances must be 2.

 

What size has the interval between ℕ*2 and ω*2?

 I take you to be asking how many ordinals are
after each of {2,4,6,...} and before ω+ω

No, the question is: What elements of {1, 2, 3, ..., ω}*2 fall between ω and ω*2?

 No doubled ordinal is between ω and ω+ω

Then {1, 2, 3, ..., ω}*2 increases the gap between {1, 2, 3, ...} and ω from 0 to infinity. Very bumpy. Only a factor 2 is possible when doubling.
Regards, WM