 Reaching this goal involves defining forty-five relations, each defined in terms of the preceding ones. These relations are all primitive  recursive.[10]  Relations needed are, among others, those which assert of a natural number that it codes a sequence, or a formula, or an axiom, or that it is the code, denoted by Sb(ru1…unZ(x1)…Z(xn)), of a formula obtained from a formula with code r by substituting for its free variable ui the xi th numeral for i = 1, …, n. The forty-fifth primitive recursive relation defined is Prf(x, y), and the forty-sixth is