Determination of the solvability of a Diophantine equation.
He immediately goes on to state that indeed the Gödel-Herbrand definition does indeed "characterize all recursive functions" – see the quote in 1934, below.
In 1930, mathematicians gathered for a mathematics meeting and retirement event for Hilbert.
As luck would have it, He announced that the answer to the first two of Hilbert's three questions of 1928 was NO.
Subsequently in 1931 Gödel published his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related To quote Kleene (1952), "The characterization of all "recursive functions" was accomplished in the definition of 'general recursive function' by Gödel 1934, who built on a suggestion of Herbrand" (Kleene 194).
Can it determine, in a finite number of steps, whether it, itself, is “successful” and "truthful" (that is, it does not get hung up in an endless "circle" or "loop", and it correctly yields a judgment "truth" or "falsehood" about its own behavior and results)?
At the 1928 Congress [in Bologna, Italy] Hilbert refines the question very carefully into three parts.Péter exhibited another example (1935) that employed Cantor's diagonal argument.Péter (1950) and Ackermann (1940) also displayed "transfinite recursions", and this led Kleene to wonder: that all "recursions" involve (i) the formal analysis he presents in his §54 Formal calculations of primitive recursive functions and, (ii) the use of mathematical induction.The debate and discovery of the meaning of "computation" and "recursion" has been long and contentious.This article provides detail of that debate and discovery from Peano's axioms in 1889 through recent discussion of the meaning of "axiom".By 1922, the specific question of an "Entscheidungsproblem" applied to Diophantine equations had developed into the more general question about a "decision method" for any mathematical formula.Martin Davis explains it this way: Suppose we are given a "calculational procedure" that consists of (1) a set of axioms and (2) a logical conclusion written in first-order logic, that is—written in what Davis calls "Frege's rules of deduction" (or the modern equivalent of Boolean logic).In 1889, Giuseppe Peano presented his The principles of arithmetic, presented by a new method, based on the work of Dedekind.Soare proposes that the origination of "primitive recursion" began formally with the axioms of Peano, although "Well before the nineteenth century mathematicians used the principle of defining a function by induction."In principle, an algorithm for [the] Entscheidungsproblem would have reduced all human deductive reasoning to brute calculation"." ...it seemed clear to Hilbert that with the solution of this problem, the Entscheidungsproblem, that it should be possible in principle to settle all mathematical questions in a purely mechanical manner.