 The many-one degrees \(\mathcal{D}_m\) and the Turing degrees \(\mathcal{D}_T\) are sometimes referred to as hierarchies in the sense that they determine an ordering on \(\mathcal{P}(\mathbb{N})\)—i.e., the set of subsets of the natural numbers—in terms of relative computability. In a series of papers from the 1940s and 1950s, Kleene (initiating in 1943) and Mostowski (initiating in 1947) realized that it was also possible to impose another sort of ordering on \(\mathcal{P}(\mathbb{N})\) in terms of the logical complexity of the simplest predicate which defines a set \(A \subseteq \mathbb{N}\) in the languages of first- or second-order arithmetic. This idea leads to what are known as the arithmetical and analytical hierarchies, both of which can be understood as classifying sets in terms of their definitional (or descriptive) complexity. As we will see, the resulting classifications are related to those determined relative to \(\mathcal{D}_T\) in terms of relative computability. They are also similar in form to other definability hierarchies studied in  computational complexity theory  (e.g., the polynomial hierarchy) and  descriptive set theory  (e.g., the Borel and projective hierarchies).