 One way to implement the B-H-K explanation of intuitionistic truth for arithmetic is to associate with each sentence \(E\) of \(\mathbf{HA}\) some collection of numerical codes for algorithms which could establish the constructive truth of \(E.\) Following Kleene [1945], a number \(e\) realizes a sentence \(E\) of the language of arithmetic by induction on the logical form of \(E\): 