Re: The set of necessary FISONs

On 06.02.2025 15:57, Jim Burns wrote:

On 2/6/2025 3:54 AM, WM wrote:

The axiom of induction:
∀P:ℕ₁→{⊤,⊥}:

That appears like nonsense. I prefer Wikipedia:
∀P( P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)).

The key is that  ∀ᴺ¹n: ∃ᴺ¹j′: n<j′

The key is that the set ℕ is created by induction.
If the set M is described as the smallest set satisfying
1 ∈ M and n ∈ M ==> n+1 ∈ M
then ℕ\M = Ø.
There is nothing remaining.

By the axiom of induction
the result is the empty set.

 You (WM) agree that  ∀ᴺ¹n: ∃ᴺ¹j′: n<j′

No. I agree to what I wrote.
Regards, WM