Definition. Let $S$ be a non-empty set, and $R$ be a binary relation on $S$ Then $R$ is said to be a well-founded relation if and only if every nonempty subset $X\subseteq S$ has an $R$ minimal element. When $R$ is well-founded, we also call the underlying set $S$ well-founded.

Note that $R$ is by no means required to be a total order, or even a partial order. When $R$ is a partial order, then $R$ minimality is the same as minimality (of the partial order). A classical example of a well-founded set that is not totally ordered is the set $\Nat$ of natural numbers ordered by division, i.e. $aRb$ if and only if $a$ divides $b$ and $a\not=1$ The $R$ minimal elements of $\Nat$ are the prime numbers.

Let $\Phi$ be a property defined on a well-founded set $S$ The principle of well-founded induction states that if the following is true :

1. $\Phi$ is true for all the $R$ minimal elements of $S$
2. for every $a$ if for every $x$ such that $xRa$ we have $\Phi(x)$ then we have $\Phi(a)$

As an example of application of this principle, we mention the proof of the fundamental theorem of arithmetic : every natural number has a unique factorization into prime numbers. The proof goes by well-founded induction in the set $\Nat$ ordered by division.