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

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.14$