Spacetime singularities and incompleteness: epistemic and ontological remarks
Gustavo E. Romero
TL;DR
The paper reframes Penrose's singularity theorem as an incompleteness result for spacetime models in General Relativity, not as evidence for ontological singular entities. It formalizes GR with an axiomatic, semantic-analytic approach and juxtaposes Penrose with Gödel's incompleteness theorems, highlighting intrinsic limits of formal representations in both physics and mathematics. By analyzing the roles of energy conditions, global causal structure, and geodesic incompleteness, the work argues that singularities signal the breakdown of GR rather than the existence of physical objects, suggesting the need for a more fundamental theory such as quantum gravity. The study thereby clarifies epistemic boundaries, preserves predictive power within applicable domains, and points toward deeper, open-ended foundational research.
Abstract
I argue that spacetime singularities entail no ontological commitment to material entities. First, I show that Penrose's singularity theorem is best understood as a theorem of incompleteness, it demonstrates the failure of specific spacetime models within General Relativity (or any theory incorporating the Raychaudhuri equation) under certain general conditions. Although this has been done before, I adopt a novel approach based on differentiating between physical and purely formal assumptions in the axiomatic foundation of general relativity. Next, I compare Penrose's result with Gödel's incompleteness theorem, highlighting key similarities and differences. Finally, I draw philosophical conclusions regarding the limits and prospects of our epistemic reconstructions of the physical world.