Definition (Hierarchy). We call an assignment $\alpha \mapsto Z_{\alpha }$ a hierarchy if

• (i) $Z_{\alpha }$ is a transitive set
• (ii) $\mathrm{Ord}\cap Z_{\alpha }=\alpha$
• (iii) $\alpha <\beta ⟹Z_{\alpha }\subseteq Z_{\beta }$
• (iv) $\lambda \text{ limit}⟹Z_{\lambda }={\bigcup }_{\alpha <\lambda }{Z}_{\alpha }$

If $\{Z_{\alpha }:\alpha \in \mathrm{Ord}\}$ is a hierarchy, we can define $Z:={\bigcup }_{\alpha \in \mathrm{Ord}}{Z}_{\alpha }$. This is a proper class as $\mathrm{Ord}\subseteq Z$. We also define ${\rho }_Z(x):=\min \{\alpha :x\in Z_{\alpha }\}$, a notion of $Z$-rank.