  We present a unified theory for formal mathematical systems including
recursive systems closely related to formal grammars, including the predicate
calculus as well as a formal induction principle. We introduce recursive
systems generating the recursively enumerable relations between lists of terms,
the basic objects under consideration. A recursive system consists of axioms,
which are special quantifier-free positive horn formulas, and of specific rules
of inference. Its extension to formal mathematical systems leads to a formal
structural induction with respect to the axioms of the underlying recursive
system. This approach provides some new representation theorems without using
artificial and difficult interpretation techniques. Within this frame we will
also derive versions of G\"odel's First and Second Incompleteness Theorems for
a general class of axiomatized formal mathematical systems.
