Goedel's 1931 proof of the incompleteness theorem is rooted in Cantor's 1891 discovery of diagonalization. The historical route reflects the train of ideas that, probably, led Goedel to his proof. It also leads naturally to the proof of the fixed point theorem, first stated by Carnap in 1934. There is an obvious further link to Kleene's 1938 recursion theorem. This kind of analysis can yield new insight, and a general theorem, of which the fixed point and recursion theorems are special cases. The setup is essentially linguistic, where "language" is conceived in a broad way. It suggests new ways of applying the diagonalization technique, in non-customary cases of "language".