 Hilbert first described a logical calculus for finitary number theory including “recursion and intuitive induction for finite totalities” in (1923, 1139).[4] Although this presentation also included a discussion of definition by simultaneous recursion, a more extensive  treatment of what we would now recognize as recursion schemes is given in his well known paper “On the infinite” (1926).  This includes a discussion of what Hilbert calls ordinary recursion (which is similar to Skolem’s description of primitive recursion), transfinite recursion, as well as recursion at higher types.   This treatment makes clear that Hilbert and his collaborators had taken substantial steps towards developing a general theory of recursive definability.  Ultimately, however, the influence of Hilbert’s presentations was diminished in light of the more precise formulation of primitive recursion which Gödel would soon provide.[5] 