Re: The set of necessary FISONs

On 2/9/2025 5:59 AM, WM wrote:

On 09.02.2025 01:00, Jim Burns wrote:

On 2/8/2025 4:54 PM, WM wrote:

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

>
Not for all inductive sets.
For all minimal.inductive sets.

>
May be called so.

Whatever the set is called,

(P(1) /\ ∀k(P(k) ==> P(k+1)),

describes each of its inductive subsets.
In a minimal.inductive set,
there is only one subset it describes,
the subset which is the whole set.

If any property or predicate P satisfies
(P(1) /\ ∀k(P(k) ==> P(k+1)),
then it describes all elements of
[a minimal.inductive] set.

0∈I ∧ ∀k:k∈I⇒k+1∈I
I is inductive.
0∈M ∧ ∀k:k∈M⇒k+1∈M
0∈S ∧ ∀k:k∈S⇒k+1∈S ∧ S⊆M ⇒ S=M
M is minimal.inductive.

minimal.inductive ≠ inductive ≠ infinite

>
Then you are wrong.
Every inductive set is infinite.

Some infinite sets, such as E(137),
are not inductive.
Also,
some inductive sets, such as ℝ,
are not minimal.inductive.
ℝ is inductive
0∈ℝ ∧ ∀k:k∈ℝ⇒k+1∈ℝ
ℝ is not minimal.inductive
0∈ℤ ∧ ∀k:k∈ℤ⇒k+1∈ℤ ∧ ℤ⊆ℝ ⇒⃒ ℤ=ℝ