Re: The set of necessary FISONs

On 2/27/2025 2:50 PM, WM wrote:

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.

<WM>

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

</WM>
Date: Thu, 6 Feb 2025 17:55:57 +0100
Message-ID: <vo2pit$31hlr$1@dont-email.me>

If A(n) is useless for UA = ℕ,
then A(n+1) us useless too.
No reason to extend this simple concept.

You extend ∀n:Aᴺ(n) to Aᴺ(ℕ)
but you only claim it, you don't justify it.

I do it order to avoid the following waffle:

How very Orwellian of you.
I justify my claims. Doing so is 'waffle'.
You don't. Abstaining from doing so is
'mathematics' and 'logic' and 'geometry'.

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.

Zermello's Infinity guarantees a superset Z of Z₀
 From Z, it follows,
by Powerset and by Separation,
that Z₀ exists.
∀n:Aᴺ(n) ⇒ Aᴺ(ℕ)
is your fantasy.
You would find your posts greatly improved
by criticizing (if you can) _our_ reasoning,
instead of
whatever you (WM) have mistaken for our reasoning.

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 we have shown that there is
the intersection of all inductive subsets of
an inductive set,
then we have constructed ℕ.
In this context,
a 'construction' is a proof of existence.

When all n are subtrated  by induction from ℕ
then we have created the empty set.
Do you agree?

I am trying to reach some expressions
with which I can answer you and be understood.
I'm not there yet.