Re: how

On 6/22/2024 2:16 PM, Chris M. Thomasson wrote:

On 6/22/2024 1:23 PM, WM wrote:

Le 22/06/2024 à 21:09, "Chris M. Thomasson" a écrit :

On 6/22/2024 5:11 AM, WM wrote:

>

∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
That statement covers all FISONs.

>
An infinite set of FISON's? Sorry for my notation here, but it should get the point across:
>
[ { 1 }, { 1, 2 }, { 1, 2, 3 }, { 1, 2, 3, 4}, ... ]
>
?
>

Every existing set of ordinals has a first or smallest element. The set of FISONs necessary to yield ℕ would be a set of ordinals. It is well defined since for every FISON we can determine whether it is necessary. It has no smallest element. ==> It is empty.
>
IF ℕ is the union of FISONs, THEN ℕ is empty. ℕ is not empty.
Regards, WM
>
Regards, WM
>
>

 A infinite sequence of FISONS:
 r[0] = { 1 }
r[1] = { 1, 2 }
r[2] = { 1, 2, 3 }
r[3] = { 1, 2, 3, 4 }
...
 See?

Zero index is interesting here... ;^)
r[1] = { 1 }
r[2] = { 1, 2 }
r[3] = { 1, 2, 3 }
r[4] = { 1, 2, 3, 4 }
In this iteration, humm... How about:
r[0] = { }
r[1] = { 1 }
r[2] = { 1, 2 }
r[3] = { 1, 2, 3 }
r[4] = { 1, 2, 3, 4 }
;^D