Exercise
51
[1XY] Difficulty:*. Let \(A\) be a well-ordered set 1 by the order \(≤\); let \(m=\min A\); then for propositions \(P(a)\) with \(a∈ A\) you can use a proof method, called transfinite induction, in which
\(P(m)\) is required to be true, and
the following ”inductive step” is proven:
\[ ∀ n∈ ℕ \Big( \big(∀ k{\lt} n, P(k)\big)⇒ P(n)\Big) \]
Show that if the proposition \(P\) satisfies the previous two requirements, then \(∀ x∈ A,P(x)\).
Prove also that if \(A=ℕ\) then the ”inductive step” is equivalent to the inductive step of strong induction (defined in [1XS]).