Diagonalization, Language and Names

There is a direct route from Cantor's 1891 diagonal construction to
the 1934 Goedel-Carnap fixed point theorem, which is also probably how
Goedel got the idea of his proof. The route goes on to  Kleene's 1938
recursion theorem. When these results are viewed in the right light,
they turn out to be special cases of a general theorem, which is
phrased in terms of a generalized notion of language and naming. The
modeling I shall suggest makes place for uncustomary "languages",
and enables a uniform treatment of various results. It also leads to
some technical questions,  of which I know partial answers.
There is a philosophical aspect to all this, concerning the notion of
"language" and connections to the Liar paradox.