# Definition:Inductive Class

## Definition

Let $A$ be a class.

Then $A$ is inductive if and only if:

 $(1)$ $:$ $A$ contains the empty set: $\ds \quad \O \in A$ $(2)$ $:$ $A$ is closed under the successor mapping: $\ds \forall x:$ $\ds \paren {x \in A \implies x^+ \in A}$ where $x^+$ is the successor of $x$ That is, where $x^+ = x \cup \set x$

### Inductive Set

The same definition can be applied when $A$ is a set:

Let $S$ be a set of sets.

Then $S$ is inductive if and only if:

 $(1)$ $:$ $S$ contains the empty set: $\ds \quad \O \in S$ $(2)$ $:$ $S$ is closed under the successor mapping: $\ds \forall x:$ $\ds \paren {x \in S \implies x^+ \in S}$ where $x^+$ is the successor of $x$ That is, where $x^+ = x \cup \set x$

## General Definition

Let $A$ be a class.

Let $g: A \to A$ be a mapping on $A$.

Then $A$ is inductive under $g$ if and only if:

 $(1)$ $:$ $A$ contains the empty set: $\ds \quad \O \in A$ $(2)$ $:$ $A$ is closed under $g$: $\ds \forall x:$ $\ds \paren {x \in A \implies \map g x \in A}$

## Also see

• Results about inductive classes can be found here.