Re: how

On 6/22/2024 4:23 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 [nonempty] 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.

IF ℕ is the union of necessary FISONs,
THEN ℕ is empty.
ℕ is not empty.
ℕ is not the union of necessary FISONs,
⎛ No single FISON is necessary.
⎜ Even! No _finite_ FISON.subset is necessary.
⎝ The whole infinite FISON.set is necessary.
The FISON.union is the FISON.union.