The quest for complete observables in general relativity has been a long-standing open problem. We employ methods from descriptive set theory to show that no complete observable on rich enough collections of spacetimes is Borel definable. In fact, we show that it is consistent with the Zermelo-Fraenkel and dependent choice axioms that no complete observable for rich collections of spacetimes exists whatsoever. In a nutshell, this implies that the problem of observables is to “analysis” what the Delian problem was to “straightedge and compass.” Our results remain true even after restricting the space of solutions to vacuum solutions. In other words, the issue can be traced to the presence of local degrees of freedom. We discuss the next steps in a research program that aims to further uncover this novel connection between theoretical physics and descriptive set theory.

• Received 9 May 2023
• Revised 23 August 2023
• Accepted 29 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.171402