Re: how

On 4/13/24 8:35 AM, WM wrote:

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

ALL elements of ℕ are individually usable, there is not finite number n that is the highest usable number, which is the fundamental FLAW in your logic, which makes it just BLOW UP.

 

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.

NONE.
All of the elements of { 1, 2, 3, ...} *2 will be < ω, and the {ω}*2 part will BE ω*2, and thus not between.

>

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?

None.

>
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.

Nope, because the "gap" doesn't have a defined distance, just an ordering.

 Regards, WM