One advantage of this alternative realization is that equality is identity, not an inductively defined relation.

Since actions are defined inductively, so is the function h, known as the order of the hierarchical complexity.

An -tuple is defined inductively using the construction of an ordered pair.

The satisfaction relation for path formulae and a path is also defined inductively.

Lists are defined inductively: The atom is a list.

The classifications and are defined inductively for every natural number n using the following rules:

The classifications and are defined inductively with the following rules.

The set of free variables of an expression is defined inductively:

Standard translation is defined inductively on the structure of the formula.

Modern proof theory treats proofs as inductively defined data structures.