Equivalence of Well-Ordering Principle and Induction

Theorem
The Well-Ordering Principle, the Principle of Mathematical Induction and the Principle of Complete Induction are logically equivalent.

That is:


 * Principle of Mathematical Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
 * $0 \in S$
 * $n \in S \implies n + 1 \in S$
 * then $S = \N$.




 * Principle of Complete Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
 * $0 \in S$
 * $\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$
 * then $S = \N$.




 * Well-Ordering Principle: Every nonempty subset of $\N$ has a minimal element.

Proof
To save space, we will refer to:
 * The Well-Ordering Principle as WOP
 * The Principle of Mathematical Induction as PMI
 * The Principle of Complete Induction as PCI.

Final assembly
So, we have that:
 * PMI implies PCI: The Principle of Mathematical Induction implies the Principle of Complete Induction
 * PCI implies WOP: The Principle of Complete Induction implies the Well-Ordering Principle
 * WOP implies PMI: The Well-Ordering Principle implies the Principle of Mathematical Induction.

This completes the result.