 Putting aside Gödel’s use of the term “recursive” rather than “primitive recursive” (which will be explained below), this exposition comes close to coinciding with the contemporary definition of the primitive recursive functions given in  Section 2.1.[6]  Gödel’s definition also improved upon those of his predecessors by clearly defining the class of initial functions which are allowed in primitive recursive definitions and by stating that each primitive recursive function possesses a definition in terms of a sequence of functions showing how it is built up from initial functions. This makes clear that the primitive recursive functions constitute a mathematically well-defined class of functions on the natural numbers (which will be denoted here as  \(\mathbf{PR}\)).  Gödel additionally proved that the primitive recursive relations—defined as characteristic functions via (\ref{prch})—are closed under propositional operations and quantification bounded by a primitive recursive function (see  Section 2.1.2).