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On 07.02.2025 17:10, Jim Burns wrote:So all elements are individually not needed.On 2/6/2025 2:32 PM, WM wrote:The axiom of induction says: If any property or predicate P satifies>>I prefer Wikipedia:>
∀P (P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)).
That's intended to be part of the definition of ℕ₁
As well it is
the definition of the collection of all FISONs.
Which is curious, when one considers that
the collection of all FISONs appears nowhere in it.
(P(1) /\ ∀k(P(k) ==> P(k+1)), then it describes all elements of an inductive = infinite set. That is satisfied by the set M of all FISONs which are useless in U(A(n)) = ℕ.
No, all of the FISONs can be individually discarded.But only all FISONs can be discarded.If the set M is described as the smallest set satisfying>
F(1) ∈ M and F(n) ∈ M ==> F(n+1) ∈ M
then M contains
all FISONs which can be subtracted from U(Fn))
without changing the assumed result ℕ.
Also, any superset of (emptiest) M
contains at least what M contains, and
thus also contains all FISONs.
Regards, WM
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