Let \(J\) be a well-ordered set. A subset \(J_0\) of \(J\) is said to be inductive if for every \(\alpha \in J\),
\[ (S _ {\alpha} \subset J _ {0}) \Longrightarrow \alpha \in J _ {0} \]
Theorem (The principle of transfinite induction). If \(J\) is a well-ordered set and \(J_0\) is an inductive subset of \(J\), then \(J_0 = J\) .
Solution
Suppose \(J_0\ne J\). Since \(J\) is well-ordered, the nonempty set \(J-J_0\) has a smallest element; call it \(\alpha\). Every \(x<\alpha\) lies in \(J_0\) by minimality of \(\alpha\). Hence \[ S_\alpha\subset J_0. \] Since \(J_0\) is inductive, this implies \(\alpha\in J_0\), contradicting \(\alpha\in J-J_0\). Therefore \[ \boxed{J_0=J}. \]