Post Emil L.. Finite combinatory processes—formulation 1. The journal of symbolic logic, vol. 1 (1936), pp. 103–105.

structions, can be regarded as a kind of Turing machine. I t is thus immediately clear that computability, so defined, can be identified with (especially, is no less general than) the notion of effectiveness as it appears in certain mathematical problems (various forms of the Entscheidungsproblem, various problems to find complete sets of invariants in topology, group theory, etc., and in general any problem which concerns the discovery of an algorithm). The principal result is that there exist sequences (well-defined on classical grounds) which are not computable. In particular the deducibility problem of the functional calculus of first order (Hilbert and Ackermann's engere Funktionenkalkiil) is unsolvable in the sense that, if the formulas of this calculus are enumerated in a straightforward manner, the sequence whose nth term is 0 or 1, according as the nth formula in the enumeration is or is not deducible, is not computable. (The proof here requires some correction in matters of detail.) In an appendix the author sketches a proof of equivalence of "computability" in his sense and "effective calculability" in the sense of the present reviewer {American journal of mathematics, vol. 58 (1936), pp. 345-363, see review in this JOURNAL, vol. 1, pp. 73-74). The author's result concerning the existence of uncomputable sequences was also anticipated, in terms of effective calculability, in the cited paper. His work was, however, done independently, being nearly complete and known in substance to a number of persons at the time that the paper appeared. As a matter of fact, there is involved here the equivalence of three different notions: computability by a Turing machine, general recursiveness in the sense of Herbrand-Godel-Kleene, and X-definability in the sense of Kleene and the present reviewer. Of these, the first has the advantage of making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately—i.e. without the necessity of proving preliminary theorems. The second and third have the advantage of suitability for embodiment in a system of symbolic logic.