Definition:Rank (Set Theory)

Definition
Let $A$ be a set.

Let $V$ denote the von Neumann hierarchy.

Then the rank of $A$ is the smallest ordinal $x$ such that $A \in \map V {x + 1}$, given that $x$ exists.

Notation
The rank of the class $A$ is sometimes denoted as $\map {\operatorname {rank} } A$.

Also see

 * Set has Rank