Definition:Inductive Set

Definition

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 $x^+ = x \cup \set x$

Axiomatic Set Theory

The concept of an inductive set is axiomatised in the Axiom of Infinity in Zermelo-Fraenkel set theory:

$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies \paren {u \cup \set u \in x} }$

Also known as

Some sources refer to this concept as a successor set.

However, note that this term has already been used on this site.

Examples

Inductive Set as Subset of Real Numbers

A specific instance of such an inductive set is defined by some authors as follows:

Let $I$ be a subset of the real numbers $\R$.

Then $I$ is an inductive set if and only if:

$1 \in I$

and

$x \in I \implies \paren {x + 1} \in I$

Also see

• Definition:Inductive Class, of which this is an instance

• Results about inductive sets can be found here.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Infinity