Re: The set of necessary FISONs

On 2/20/2025 12:50 PM, WM wrote:

On 20.02.2025 15:30, Jim Burns wrote:

AXIOM I 'extensionality' means
proving each FISON is omissible  is no
proving {F} is omissible.

>
∀n ∈ ℕ: n+1 ∈ ℕ.
Together with 1 ∈ ℕ this defines the set ℕ.
 (*)

You (WM) have left out that
ℕ is the only.inductive.subset of ℕ
S⊆ℕ ∧ 1∈S ∧ ∀n∈S:n+1∈S ⇒ S=ℕ
  (**)

Addition of all numbers defined by (*)
to the empty set
is tantamount to addition of ℕ to the empty set.

{**} prevents the addition of extra elements.

Subtraction of all numbers defined by (*)
from ℕ
is tantamount to subtraction of ℕ from ℕ.
>
Homework:
Prove the same for FISONs or v. Neumann ordinals.

(!) Have you (WM) started reading my proofs?
For the sets of all (finite) FISONs and
of all finite von Neumann ordinals,
(**) is satisfied as a consequence of
the finitude of their elements.
Finitude prevents darkness (extra.ness).