 The foregoing sequence of definitions provides some evidence for the robustness of the class of primitive recursive relations and functions. Further evidence is provided by the fact that it is possible to develop the machinery for coding and decoding finite sequences of natural numbers and for performing various combinatorial operations on sequences—e.g., adjunction of an element, concatenation, extracting a subsequence, substituting one element for another, etc. The primitive recursiveness of these operations underpins Gödel’s arithmetization of syntax as described in  Section 1.3.  We present here only the basic definitions required to demonstrate the primitive recursiveness of the \(k\)-tupling and projection functions which are required for results in computability theory such as the Normal Form Theorem  (2.3)  discussed below.