Re: The set of necessary FISONs

On 2/7/2025 11:39 AM, WM wrote:

On 07.02.2025 17:10, Jim Burns wrote:

On 2/6/2025 2:32 PM, WM wrote:

On 06.02.2025 19:54, Jim Burns wrote:

On 2/6/2025 11:55 AM, WM wrote:

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

>
That's intended to be part of the definition of ℕ₁

>
As well it is
the definition of the collection of all FISONs.

>
Which is curious, when one considers that
the collection of all FISONs appears nowhere in it.

>
The axiom of induction says:
If any property or predicate P satifies
(P(1) /\ ∀k(P(k) ==> P(k+1)),
then it describes all elements of
an inductive = infinite set.

All inductive sets are infinite.
Not all infinite sets are inductive.
Not all inductive sets 𝕂
  0∈𝕂 ∧ ∀k:k∈𝕂⇒k+1∈𝕂
can use induction validly (that is: without exception)
  ∀P∈{T,F}ᵂ:( P(0) ∧ ∀k∈𝕎:P(k)⇒P(k+1) ⇒ ∀n∈𝕎:P(n) )
Yes, 'Inductive set' sounds like
it should mean that, but it doesn't.
I'd prefer 'pre.inductive', but no one asked me.
However,
a minimal.inductive set 𝕄
  𝕄 inductive ∧ 𝕄 ⊃≠ S not.inductive
can always validly use induction.
  ∀P∈{T,F}ᵂ:( P(0) ∧ ∀k∈𝕎:P(k)⇒P(k+1) ⇒ ∀n∈𝕎:P(n) )
is essentially the claim that 𝕎 is minimal.inductive.
⎛ Assume otherwise.
⎜ Assume 𝕎 is inductive and
⎜ its proper.subset 𝕍 is inductive.
⎜
⎜ Induction gives an incorrect result
⎜ for at least one predicate P(n) :⇔ n ∈ 𝕍
⎜ The conditions for induction are satisfied
⎜⎛ P(0)
⎜⎜ ∀k∈𝕍:P(k)⇒P(k+1)
⎜⎝ ∀k∈𝕎\𝕍:¬P(k)∨P(k+1)
⎜ but
⎝ ¬∀n∈𝕎:P(n)
An inductive set which can validly use induction
does not have an inductive proper.subset.

That is satisfied by the set M of all FISONs which are useless in U(A(n)) = ℕ.

Only not.followed FISONs are not (your term) uselessᵂᴹ.
Each FISON is followed.
{F:uselessᵂᴹ} = {F}