Resolving G\"odel's Incompleteness Myth: Polynomial Equations and Dynamical Systems for Algebraic Logic

TL;DR

This paper introduces a novel computational approach using polynomial equations and dynamical systems to evaluate logical propositions, challenging traditional interpretations of G"odel's incompleteness theorems and providing a new perspective on undecidability.

Contribution

It presents a new algebraic and dynamical systems framework for analyzing logical formulas, offering insights into G"odel's theorems and the nature of undecidability in mathematics.

Findings

Logical formulas can be represented as polynomial equations and dynamical systems.

G"odel's self-referential formula is shown to be inconsistent or unsteady within this framework.

The approach clarifies that G"odel's incompleteness does not imply fundamental limitations of mathematics.

Abstract

A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms is computed from the solution to the corresponding system of polynomial equations. A reference by a formula to its own provability is shown to be a recurrence relation, which can be either interpreted as such to generate a discrete dynamical system, or interpreted in a static way to create an additional simultaneous equation. In this framework the truth values of logical formulas and other polynomial objectives have complex data structures: sets of elementary values, or dynamical systems that generate sets of infinite sequences of such solution-value sets. Besides the routine result that a formula has a definite elementary value, these data structures encode several exceptions: formulas that are ambiguous, unsatisfiable, unsteady, or contingent. These exceptions represent several semantically different types of undecidability; none causes any fundamental problem for mathematics. It is simple to calculate that Goedel's formula, which asserts that it cannot be proven, is exceptional in specific ways: interpreted statically, the formula defines an inconsistent system of equations (thus it is called unsatisfiable); interpreted dynamically, it defines a dynamical system that has a periodic orbit and no fixed point (thus it is called unsteady). These exceptions are not catastrophic failures of logic; they are accurate mathematical descriptions of Goedel's self-referential construction. Goedel's analysis does not reveal any essential incompleteness in formal reasoning systems, nor any barrier to proving the consistency of such systems by ordinary mathematical means.