Implications of Tarski's Undefinability Theorem on the Theory of Everything

TL;DR

This paper explores how Tarski's undefinability theorem constrains formal theories of everything and discusses the potential for non-algorithmic understanding to overcome these limitations.

Contribution

It proposes axioms for a theory of everything and analyzes the impact of Tarski's theorem on the definability of scientific truths within such theories.

Findings

Tarski's theorem limits formal systems in defining scientific truths.

Non-algorithmic understanding may transcend formal limitations.

Implications for the foundational limits of a unified physical theory.

Abstract

The Theory of Everything ($S_{\text{ToE}}$) seeks to unify all fundamental forces of nature, including quantum gravity, into a single theoretical framework. This theory would be defined internally using a set of axioms, and this paper proposes a set of axioms for any such theory. Furthermore, for such a theory, all scientific truth would be defined internally as consequences derivable from the rules of such a theory. This paper then examines the implications of Tarski's undefinability theorem on scientific truths derived from such axioms. We demonstrate that Tarski's theorem imposes limitations on any such formal system $S_{\text{ToE}}$. However, we also argue that the Lucas-Penrose argument suggests that non-algorithmic understanding can transcend these formal limitations.