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).