# Logical Dependence of Induction on the Well-Ordering Principle

I know from a Discrete Mathematics class in the spring that Mathematical Induction depends on the well-ordering principle for natural numbers. The explanation in my textbook (Rosen) did not give me the level of understanding I was hoping for.

Could someone explain the equivalence of the well-ordering principle and the principle of mathematical induction?

I am aware there are resources for proofs of this, but I am looking for a more "intuitive" explanation than a proof- proofs I can read on my own.

• Induction works from zero "upwards", showing that every natural number satisfies some property. You first show that zero has the property (the base case), and then that if a number $n$ has the property than so does $n+1$ (the successor step).
• The well ordering principle works "downwards", but to prove the equivalence with induction you look at the least number that does not satisfy the property. If you look at the least number that does not satisfy the property, then either that number is zero (which is ruled out by the base case of the induction proof) or that number is of the form $n+1$ (which is ruled out by the successor step of the induction proof).