The principle of complete induction is not only valid for statements about natural numbers but for statements about elements of any well-founded set, that is, a set with an irreflexive relation < that contains no infinite descending chains. Any set of cardinal numbers is well-founded, which includes the set of natural numbers.

Applied to a well-founded set, it can be formulated as a single step:

- Show that if some statement holds for all
*m*<*n*, then the same statement also holds for*n*.

This form of induction, when applied to a set of ordinals (which form a well-ordered and hence well-founded class), is called *transfinite induction*. It is an important proof technique in set theory, topology and other fields.

Proofs by transfinite induction typically distinguish three cases:

- when
*n*is a minimal element, i.e. there is no element smaller than*n*; - when
*n*has a direct predecessor, i.e. the set of elements which are smaller than*n*has a largest element; - when
*n*has no direct predecessor, i.e.*n*is a so-called limit ordinal.

Strictly speaking, it is not necessary in transfinite induction to prove a base case, because it is a vacuous special case of the proposition that if *P* is true of all *n* < *m*, then *P* is true of *m*. It is vacuously true p

The principle of complete induction is not only valid for statements about natural numbers but for statements about elements of any well-founded set, that is, a set with an irreflexive relation < that contains no infinite descending chains. Any set of cardinal numbers is well-founded, which includes the set of natural numbers.

Applied to a well-founded set, it can be formulated as a single step:

- Show that if some statement holds for all
*m*<*n*, then the same statement also holds for*n*.

This form of induction, when applied to a set of ordinals (which form a well-ordered and hence well-founded class), is called *ordinals (which form a well-ordered and hence well-founded class), is called transfinite induction. It is an important proof technique in set theory, topology and other fields.
*

*Proofs by transfinite induction typically distinguish three cases:
*

- when
*n*is a minimal element, i.e. there is no element smaller than*n<*Proofs by transfinite induction typically distinguish three cases:

Strictly speaking, it is not necessary in transfinite induction to prove a base case, because it is a vacuous special case of the proposition that if

*P*is true of all*n*<*m*, then*P*is true of*m*. It is vacuously true precisely because there are no values of*n*<*m*that could serve as counterexamples. So the special cases are special cases of the general case.## Relationship to the well-ordering principle