Formal Mathematical Systems including a Structural Induction Principle

Abstract

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.