Definition:Infinite Successor Set

Definition

An infinite successor set is a set $S$ defined as:

$\varnothing \in S$

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

where $x^+$ 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:

$\exists x: \left({\left({\exists y: y \in x \land \forall z: \neg \left({z \in y}\right)}\right) \land \forall u: u \in x \implies \left({u \cup \left\{{u}\right\} \in x}\right)}\right)$

Also known as

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

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 11$: Numbers