Nonmonotone Inductive Definitions

We have seen in Sect. 6.3 that the fixed-point of an inductive definition comes in stages. We used a cardinality argument to show that the hierarchy of stages of an inductive definition becomes eventually stationary. Monotonicity, however, is not really necessary in the cardinality argument. The cardinality argument only needs that the operator is inflationary, i.e., that the operator Φsatisfies the condition X ⊆ Φ(X). This means, however, not really a restriction since any operator Φ:Pow(N) → Pow(N) induces an operator Φ′(X) :=X∪Φ(X) which is apparently inflationary.