left and right euclidean;
Given a subset of , suppose that has the property that, given any element of , if
then . Such an may be called a -inductive subset of . The relation is well-founded if the only -inductive subset of is itself.
Note that this is precisely what is necessary to validate induction over : if we can show that a statement is true of whenever it is true of everything -below , then it must be true of everything in . In the presence of excluded middle it is equivalent to other commonly stated definitions; see Formulations in classical logic below.
While the definition above follows how a well-founded relation is generally used (namely, to prove properties of elements of by induction), it is complicated. Two alternative formulations are given by the following:
The relation is classically well-founded if every inhabited subset of has a member such that no satisfies . (Such an is called a minimal element of .)
In classical mathematics, both of these conditions are equivalent to being well-founded. Constructively, we may prove that a well-founded relation has no infinite descent, but not the converse; and we may prove that a classically well-founded relation is well-founded, but not the converse. (In fact, if there exists an inhabited relation that is classically well-founded, then excluded middle follows.) In predicative mathematics, however, the definition of well-founded may be impossible to even state, and so either of these alternative definitions would be preferable (if classical logic is used).
Even in constructive predicative mathematics, (1) is strong enough to establish the Burali-Forti paradox (when applied to linear orders). In material set theory, (2) is traditionally used to state the axiom of foundation, although the impredicative definition could also be used as an axiom scheme (and must be in constructive versions). In any case, either (1) or (2) is usually preferred by classical mathematicians as simpler.
Many inductive or recursive notions may also be packaged in coalgebraic terms. For the concept of well-founded relation, first observe that a binary relation on a set is the same as a coalgebra structure for the covariant power-set endofunctor on , where if and only if .
the map factors through . (Note that is necessarily monic, since preserves monos.) Unpacking this a bit: for any , if belongs to , that is if , then . This says the same thing as .
Then, as usual, the -coalgebra is well-founded if every -inductive subset is all of .
In coalgebraic language, a simulation is simply a -coalgebra homomorphism . Condition (1), that is merely -preserving, translates to the condition that is a colax morphism of coalgebras, in the sense that there is an inclusion
Every well-founded relation is irreflexive; that is, . Sometimes one wants a reflexive version of a well-founded relation; let if and only or . Then the requirement that be a minimal element of a subset states that only if . But infinite descent or direct proof by induction still require rather than .
The axiom of foundation in material set theory states precisely that the membership relation on the proper class of all pure sets is well-founded. In structural set theory, accordingly, one uses well-founded relations in building structural models of well-founded pure sets.
Again let be the set of natural numbers, but now let if in the usual order. That this relation is well-founded is the principle of strong induction.
More generally, let be a set of ordinal numbers (or even the proper class of all ordinal numbers), and let if in the usual order. That this relation is well-founded is the principle of transfinite induction.
Let be the set of integers, and let mean that properly divides : is an integer other than . This relation is also well-founded, so one can prove properties of integers by induction on their proper divisors.