Re: The set of necessary FISONs

On 20.02.2025 20:46, Jim Burns wrote:

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

ℕ_def to be precise.

 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.

No extra elements are available and no extra elements shall be added.

 

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?

Why should I? I discuss my proof.

 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.

S = ℕ_def.
Regards, WM