all natural numbers, and is the graph of the successor function . Then induction on is the usual mathematical induction, and recursion on gives primitive recursion. If we consider the order relation , we obtain complete induction, and course-of-values recursion. The statement that is well-founded is also known as the well-ordering principle. There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction. When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those articles for more details. Examples Well-founded relations that are not totally ordered include: The positive integers , with the order defined by if and only if divides and . The set of all finite strings over a fixed alphabet, with the order defined by if and only if is a proper substring of . The set of pairs of natural numbers, ordered by if and only if and . Every class whose elements are sets, with the relation ∈ ("is an element of"). This is the axiom of regularity. The nodes of any finite directed acyclic graph, with the relation defined such that if and only if there is an edge from to . Examples of relations that are not well-founded include: The negative integers , with the usual order, since any unbounded subset has no least element. The set of strings over a finite alphabet with more than one element, under the usual (lexicographic) order, since the sequence is an infinite descending chain. This relation fails to be well-founded even though the entire set has a minimum element, namely the empty string. The set of non-negative rational numbers (or reals) under the standard ordering, since, for example, the subset of positive rationals (or reals) lacks a minimum. Other properties If is a well-founded relation and is an element of , then the descending chains starting at are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let be the union of the positive integers with a new element ω that is bigger than any integer. Then is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length;