 Recall that according to the definitions given in  Section 3.3,  a relation \(R \subseteq \mathbb{N}^k\) is said to be computable just in case its characteristic function \(\chi_R(\vec{x})\) is a computable function and computably enumerable just in case it is the range of a computable function. In order to introduce the arithmetical hierarchy, it is useful to employ an alternative characterization of computable and computably enumerable relations in the form of a semantic analog to the proof-theoretic notion of arithmetical representability discussed in  Section 1.3.