# Definition:Minimal Infinite Successor Set/Definition 1

## Definition

Let $S$ be an infinite successor set.

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

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

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

## Also see

• Results about the minimal infinite successor set can be found here.