# 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 $A \in V \left({x+1}\right)$, given that $x$ exists.

## Notation

The rank of the class $A$ is sometimes denoted as $\operatorname{rank} \left({A}\right)$.