Emil L. Post was very close to proving Gödel's incompleteness theorem, and the existence of algorithmically unsolvable problems in the early 1920s. He realized that one could enumerate all algorithms, and hence obtain an unsolvable problem by diagonalization. Moreover, the "problem" can be viewed as a computable list of questions $Q_1,Q_2,Q_3,\ldots$ for which the sequence of answers (yes or no) is not computable. It follows that there cannot be any complete formal system that proves all true sentences of the form "The answer to $Q_i$ is yes" or "The answer to $Q_i$ is no," because this would solve the unsolvable problem.
But then Post was stuck because he needed to formalize the notion of computation. He had in fact had (an equivalent of) the right definition, but logicians were not ready for a definition of computation, and did believe there was such a thing until the Turing machine concept came along in 1936. Gödel avoided this problem when he proved his theorem (1930) by proving incompleteness of a particular system (Principia Mathematica).