The quest for complete observables in general relativity has been a longstanding open problem. We employ methods from descriptive set theory to show that no complete observable is Borel definable. In fact, we show that it is consistent with the Zermelo-Fraenkel and Dependent Choice axioms that no complete observable 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 in general relativity.