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. In constructive mathematics, we may prove that a well-founded relation has no infinite descent (see Proposition 1), but not the converse, and that a classically well-founded relation is well-founded (see Proposition 2), but not the converse.
We note that classical well-foundedness is really too strong for constructive (i.e., intuitionistic) mathematics: if there exists an inhabited relation that is classically well-founded, then excluded middle follows. (This holds true in any topos; for a proof, see here.) On the other hand, the infinite descent condition is too weak to be of much use in constructive mathematics. It is the inductive notion of well-foundedness that is just right.
Note however that in predicative mathematics, the definition of well-founded may be impossible to even state, and so either of these alternative definitions would be preferable, provided 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.
To round out the discussion we prove the following two results, both valid in intuitionistic mathematics:
If is a well-founded relation and has no minimal element, then is empty.
This result makes it trivial to infer (under classical logic) that classical well-foundedness is a consequence of well-foundedness. It also shows that well-foundedness rules out infinite descent (intuitionistically), since an infinite descent sequence has no minimal element.
Let . Clearly . We show is inductive, so that under well-foundedness and , as desired.
So, suppose is an element such that whenever . We must show . Claim: . For if , then would be a minimal element of (as ). But this negates the assumption that has no minimal element.
Thus , and , so that . This completes the proof.
In intuitionistic set theory, classical well-foundedness implies (inductive) well-foundedness.
Let be a classically well-founded relation on , and let be an inductive subset. We must show that every element belongs to . Since is inductive, it suffices to show that every belongs to , i.e. we may assume given a such that and show . But under this assumption we have that is inhabited, so according to Remark 1, the law of excluded middle follows and we might as well then argue classically. The argument is well-known but we include it for completeness: under classical well-foundedness, if an inductive subset is not the entirety of , then the complement has a minimal element . In that case, implies , but then since is inductive, contradiction. Hence and in particular , which is what we wanted.
To bring us full circle: in classical set theory we may prove that if has no infinite descent, then is classically well-founded. For suppose an inhabited subset (say with an element ) failed to have a least element. Then we can find an infinite descent sequence with , by choosing at each stage such that . Technically this requires the use of dependent choice, but generally this is felt to be a mild choice principle (that is true even for intuitionistic mathematics).
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 .
given , there exists with .
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
If is well-founded and is (weakly) extensional, then there is at most one simulation .
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.