Re: The set of necessary FISONs

On 27.02.2025 19:19, Jim Burns wrote:

On 2/27/2025 5:45 AM, WM wrote:

On 26.02.2025 23:17, Jim Burns wrote:

 

This next bit you (WM) might like, for a change.
It looks like the pseudo.induction.rule which
you have been trying to use.

>
It is induction.

 This is what you (WM) have called induction:
⎛ Each inductive predicate A

No, I call induction a very restricted number of predicates.
If A(n) is useless for UA = ℕ, then A(n+1) us useless too. No reason to extend this simple concept. I do it order to avoid the following waffle:

What that version of 'induction' seems to say
is false if it's read literally.
It's false that
each inductive predicate is true.without.exception
_in each domain without exception
Z₀ is a subset of a set Z holding 0 and all the {a}

By the same induction
I prove UF = ℕ  ==> Ø = ℕ.

 What you use to prove that is
∀n:Aᴺ(n) ⇒ A(ℕ)

That is how Zermelo guarantees Z₀.

 That's not induction.
It seems to follow from confusion over
the difference between a set and its elements.
 

There is no difference in some cases like these:
When all n are added by induction to the empty set, then we have constructed ℕ.
When all n are subtrated  by induction from ℕ then we have created the empty set.
Do you agree?
Regards, WM