Incompleteness Theorems for Observables in General Relativity.

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.