Re: The set of necessary FISONs

On 2/18/2025 10:22 AM, WM wrote:

On 18.02.2025 13:25, Richard Damon wrote:

On 2/18/25 4:53 AM, WM wrote:

On 18.02.2025 04:02, Richard Damon wrote:

On 2/17/25 2:25 PM, WM wrote:

A set without elements is an empty set
and not capable of producing ℕ.

⋃{} ≠ ℕ

But an empty set of REQUIRED elements

required F′  ⇔  ∀S ⊆ {F}: ⋃S = ℕ  ⇒  S ∋ F′

doesn't mean we can't have a set of sufficient elements.

sufficient F′  ⇔  ∃S ⊆ {F}: ⋃S = ℕ  ∧  S ∋ F′

There is no element that could be
a meaningful member of any sufficient set.

¬∃F′: ∃S ⊆ {F}: ⋃S = ℕ  ∧  ⋃Sᐠꟳᣟ ≠ ℕ

Therefore there is no sufficient set.

You (WM) have located a problem.
You try to work around it by not.mentioning it.
What you're not.mentioning is your assumption
that none of these sets are infinite.
It's a problem, that assumption of yours.
But it's not our problem.

Therefore there is no sufficient set.

⋃{F} = ℕ
That is in conflict with your assumption
that no set is infinite.
That is not our assumption.
That is not our problem.

Of course there are,
its just they are not individually needed,
but are collectively sufficient.

>
For every FISON there is the question:
Can it belong to a collectively sufficient set.

For every FISON, the answer is yes.
⋃{F} = ℕ

For every FISON the answer is no.

{F} isn't finite.
⋃{F} = ℕ