Re: The set of necessary FISONs

On 2/23/2025 2:32 PM, WM wrote:

On 23.02.2025 19:34, Jim Burns wrote:

On 2/23/2025 9:43 AM, WM wrote:

There is no reason to consider {{F}} at all.

>
There is reason, but
only for people wanting to be correct.

>
Peano, Zermelo, or v. Neumann

...agree that {{F}} ≠ {F}

Peano, Zermelo, or v. Neumann create ℕ

Peano, Zermelo, and v. Neumann assert axioms
from which the existence of ℕ follows
in a finite.sequence of not.first.false claims.
That is what "create" means in this context.
Mathematical rebar is not welded.
Mathematical concrete is not poured.
Instead, a proof of existence is given.

as well as the set F of all FISONs
by induction over the members
for use in set theory
without being what you erroneously call correct.

A proof.by.induction shows that
some set,
such as the set {x:A(x)} of x such that A(x),
is inductive.
The conclusion of a proof.by.induction
is that {x:A(x)} is the whole set.
However,
not just any "whole set" is reliable here.
It must be a whole set such that
knowing {x:A(x)} is inductive
narrows
which set {x:A(x)} can be
to one set: that whole set.
Concluding an inductive subset is the whole set
is certain for a whole set which
is its.own.only.inductive.subset
but is NOT certain for any other whole set.

We omit all F(n) which amounts to remove F.
Like all natural numbers amount to ℕ (not {ℕ})

Each natural number is in the domain of ST+F
ℕ is not in the domain of ST+F
Induction on the natural numbers
is valid in ST+F
(using "k is a natural number", not "k ∈ N")
ST:
⎛ {} exists. X∪{y} exists.
⎝ Two same.membered sets are one set.
F:
⎛ Each set only has
⎜ emptier.by.one.subsets smaller.
⎝ (Each set is finite.)