Formal Mathematical Systems including a Structural Induction Principle

TL;DR

This paper develops a unified framework for formal mathematical systems that incorporate recursive systems, predicate calculus, and a structural induction principle, offering new representation theorems and generalized incompleteness results.

Contribution

It introduces a comprehensive theory combining recursive systems and formal induction, leading to novel representation theorems and extensions of Gödel's incompleteness theorems.

Findings

Develops a unified theory for formal systems including recursive and predicate calculus.

Establishes a formal structural induction principle for these systems.

Derives generalized versions of Gödel's Incompleteness Theorems.

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.