Consequences of Undecidability in Physics on the Theory of Everything

TL;DR

This paper explores how fundamental logical limits, like undecidability, constrain the possibility of a complete algorithmic 'Theory of Everything' in physics, suggesting some aspects of reality are inherently non-algorithmic.

Contribution

It formalizes a 'Meta-Theory of Everything' that incorporates non-algorithmic understanding to account for undecidable phenomena in physics.

Findings

Undecidable phenomena exist in fundamental physics.

A purely algorithmic 'Theory of Everything' is impossible due to logical limits.

The universe cannot be fully simulated by an algorithm.

Abstract

General relativity treats spacetime as dynamical and exhibits its breakdown at singularities. This failure is interpreted as evidence that quantum gravity is not a theory formulated within spacetime; instead, it must explain the very emergence of spacetime from deeper quantum degrees of freedom, thereby resolving singularities. Quantum gravity is therefore envisaged as an axiomatic structure, and algorithmic calculations acting on these axioms are expected to generate spacetime. However, G\"odel's incompleteness theorems, Tarski's undefinability theorem, and Chaitin's information-theoretic incompleteness establish intrinsic limits on any such algorithmic programme. Together, these results imply that a wholly algorithmic "Theory of Everything" is impossible: certain facets of reality will remain computationally undecidable and can be accessed only through non-algorithmic understanding. We formalize this by constructing a "Meta-Theory of Everything" grounded in non-algorithmic understanding, showing how it can account for undecidable phenomena and demonstrating that the breakdown of computational descriptions of nature does not entail a breakdown of science. Because any putative simulation of the universe would itself be algorithmic, this framework also implies that the universe cannot be a simulation.