 The developments just described form part of the prehistory of the subfield of  contemporary mathematical logic which was originally known as recursive function theory (or more simply as recursion theory). This subject was initiated in earnest by Kleene, Turing, and Post starting in the late 1930s, directly on the basis of the papers containing the equivalence and undecidability results summarized in  Section 1.6  and  Section 1.7.  Of particular importance are the papers (1936a, 1938, 1943, 1955a,b,c) of Kleene. These respectively contain the definition of the partial recursive functions, the proof of their equivalence to \(\mathbf{GR}\), the Normal Form Theorem, the Recursion Theorem, and the definitions of the arithmetical and analytical hierarchies. Of equal importance are the papers (1937, 1939) of Turing (which respectively contain the undecidability of the Halting Problem and the definition of Turing reducibility) and the paper (1944) of Post (which introduced many-one and one-one reducibility and formulated what would come to be known as Post’s Problem).