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On 2/21/2025 4:19 AM, WM wrote:
I have asked you to prove the same for FISONs. Exchange only ℕ by ℕ_def.Why are you asking me for proofs you won't read?>Homework:>
Prove the same for FISONs or v. Neumann ordinals.
(!) Have you (WM) started reading my proofs?
Why should I? I discuss my proof.
We need not consider {ℕ′}.ℕ_def = ℕ′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
S = {i:A(i)}
⎛ A(1) ∧ ∀n∈ℕ′:A(n)⇒A(n+1) ⇒
⎜ {i:A(i)} ∈ {S″⊆ℕ′:inductive.S″}
⎜
⎜ {S″⊆ℕ′:inductive.S″} = {ℕ′}
⎜ ⇐ all and only finites are in ℕ′
⎜
⎜ ( {ℕ′} ≠ ℕ′ )
⎜And all together are the set ℕ′.
⎜ {i:A(i)} ∈ {S″⊆ℕ′:inductive.S″} ∧
⎜ {S″⊆ℕ′:inductive.S″} = {ℕ′} ⇒
⎜ {i:A(i)} = ℕ′
⎜
⎜ {i:A(i)} = ℕ′ ∧
⎜ ∀k ∈ {i:A(i)}: A(k) ⇒
⎝ ∀k ∈ ℕ′: A(k)
Proof by induction for ℕ′
⇐ all and only finites are in ℕ′
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