# Definition:Infinite Successor Set

## Definition

Let $S$ be a set.

Then $S$ is an infinite successor set if and only if:

$\O \in S$
$x \in S \implies x^+ \in S$

where $\O$ is the empty set, and $x^+$ denotes the successor set of $x$.

### Axiomatic Set Theory

The concept of infinite successor 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

1960: Paul R. Halmos: Naive Set Theory refers to this just as a successor set, but this term has already been used on this site.