Definition:Infinite Successor Set

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Let $S$ be a set.

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

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

where $\varnothing$ 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:

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

Also known as

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.