Can quantum gravity be both consistent and complete?

TL;DR

The paper argues that achieving a complete and consistent quantum gravity may be fundamentally impossible due to Gödelian, Tarskian, and Chaitin-type limits on formal, algorithmic systems. It contrasts Universal Actualism with Selective Actualism and contends that a non-algorithmic meta-theory is indispensable for capturing truths that lie beyond computation, including the emergence of spacetime and the resolution of foundational paradoxes. By formalizing a quantum-gravity framework $\mathcal{F}_{QG}$ and demonstrating intrinsic undecidability in Planck-scale physics, it advocates a meta-theoretical structure $\mathcal{M}_{ToE}$ with an external truth predicate $T(x)$ to serve as a theory of everything. This reframes the quest for a theory of everything as inherently non-algorithmic, emphasizing information-theoretic and meta-theoretical approaches as essential components.

Abstract

General relativity, despite its profound successes, fails as a complete theory due to presence of singularities. While it is widely believed that quantum gravity has the potential to be a complete theory, in which spacetime consistently emerges from quantum degrees of freedom through computational algorithms, we argue that this goal could be fundamentally unattainable. We examine how this limitation could emerge in various contexts, depending on whether or not every mathematically valid result is physically realized. In the first case, Godel's incompleteness theorems, along with related results by Tarski and Chaitin, imply that no theory formulated as a formal axiomatic system can be complete, and that within any computational framework, a fully consistent internal truth predicate is impossible. In the second case, if only a subset of mathematical truths is realized in nature, we argue that this selection cannot be determined by any purely computational process. Hence, a meta-theoretical approach based on non-algorithmic understanding is indispensable in every case. We discuss some possible consequences of this observation for describing physical systems and note that a non-algorithmic approach should be essential for any theory of everything.