\begin{definition}[Definition:Minimal Infinite Successor Set/Definition 3]
{{Refactor|Based on Takeuti|level = advanced}}
The '''minimal infinite successor set''' $\omega$ is defined as:
:$\omega := \set {x \in \On: \paren {x \cup \set x} \subseteq K_I}$
where:
:$K_I$ is the class of all non-limit ordinals
:$\On$ is the class of all ordinals.
\end{definition}