Re: The set of necessary FISONs

On 2/15/2025 9:51 AM, WM wrote:

On 14.02.2025 19:06, Jim Burns wrote:

The set of all natural numbers is not.in
the only inductive subset of the set of
all natural numbers.

>
The set of all natural numbers is constructed
by induction.

By axiom "infinity",
an inductive set exists.
By "infinity" and "the usual suspects" set.axioms,
an inductive set with an only.inductive.subset exists.
Induction,
proving inductive the subset of those which have P, is
proving only.inductive the subset of those which have P
(though only for sets with an only.inductive subset).
⎛ The only.inductive.subset of ℕ is ℕ and
⎜ proving any subset is inductive is
⎝ proving that subset is ℕ.
If the set with an only.inductive.subset
was an element, we'd prove the set has P.
But the set isn't an element.
We don't prove the set has P.