Analytical hierarchy

Summary

The analytical hierarchy classifies sets of natural numbers by the quantifiers over sets of numbers needed to define them in second-order arithmetic. Its levels are the classes $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$. It extends the arithmetical hierarchy, whose sets all sit in $\Delta^1_1$. It is the effective, or lightface, counterpart of the projective hierarchy of descriptive set theory.