Definition:Minimally Inductive Set/Definition 1

Definition
Let $S$ be an infinite successor set.

The minimal infinite successor set $\omega$ is the infinite successor set given by:


 * $\omega := \displaystyle \bigcap \left\{{S' \subseteq S: \text{$S'$ is an infinite successor set}}\right\}$

that is, $\omega$ is the intersection of every infinite successor set which is a subset of $S$.

Also see

 * Definition:Natural Numbers as Elements of Minimal Infinite Successor Set