Abstract
This thesis has two major aims. The first is to demonstrate the centrality
of Cantor's diagonal argument in the proofs of the classical limitation
results concerning formal languages and systems - the second is to generalize
these results into a framework of more general languages and systems. The
point of generalizing the classical limitation results is to prove that
these results are in some way essential and not only results bound to languages
or systems of a very restricted type. Our concept of formal language is
defined such as to allow for instance sentences with direct self-reference
and ``poetical sentences'' that are neither true nor false. The main result
of the thesis is a version of Gödel's Incompleteness Theorem valid
for these kind of languages equipped with a suitable notion of proof. Two
smaller limitation results concerning formal languages are obtained by
a transformation of the classical semantical paradoxes of Grelling and
Epimenides into the framework of formal languages. In these transformations
the close connection between the semantical paradoxes and the diagonal
argument is revealed.