Re: The set of necessary FISONs

On 2/5/2025 4:14 AM, WM wrote:

On 04.02.2025 18:51, Jim Burns wrote:

On 2/4/2025 5:11 AM, WM wrote:

If ℕ is existing,
we do not need axioms.

>
If ℕ is described,
its description can have appended to it
not.first.false claims,

>
Either they did exist or they did not.

Either way, the axioms do not create,
but only describe.
Consider the cases.
Case 1.
The axioms are all true. ℕ exists.
⎛
⎜ The axioms can have appended to them
⎜ not.first.false claims
⎜ which we can see₀ are not.first.false.
⎜ We see₀ Q(k) is not.first.false
⎜ in ⟨ P(k) P(k)⇒Q(k) Q(k) ⟩
⎜
⎜⎛ 'See₀' is the non.metaphorical sense of 'see'.
⎜⎜ Seeing₀ is eye.work. You see₀ this sentence
⎜⎝ on a screen of some kind. You parrot sees₀ it, too.
⎜
⎜ In a finite sequence which
⎜ we see₀ are only not.first.false claims,
⎜ we know that each claim is true.
⎜ We can non.metaphorically, non.exaggeratedly
⎜ say that there, seeing₀ those claims,
⎜ we see₀ truth,
⎜ true before and after it's seen₀,
⎜ true despite any rejection of it.
⎜
⎜ Seeing₀ claims is so ordinary that
⎜ a mind might boggle at the suggestion that
⎜ such an extraordinary a thing happened:
⎜ this abstract property truth was made visible₀
⎝ Nonetheless.
Case 2.
The axioms aren't all true. ℕ doesn't exist.
⎛
⎜ Append the claims. Or don't.
⎜ The claims don't matter.
⎜ Like "The current king of France is bald",
⎜ they avoid leading us wrong,
⎜ but only because they don't refer.
⎝ They never lead us at all.

If ℕ is existing,
we do not need axioms.

>
If the axioms are contradictory,
ℕ is not existing.

>
And if they are not contradictory,
then the ℕ created by them exists.

Axioms do not change whether ℕ exists.
Axioms are useful for other reasons.

If ℕ is existing,
we do not need axioms.

>
Describing it did not create it.

>
Then nothing further than stating ℕ
would be necessary.

Necessary for what?

That is a wrong opinion.

Are there finiteᵂᴹ sequences of claims which
don't hold a first.false claim, but
hold a false claim?