Hypergraph C*-algebras are C*-algebras presented by a finite set of projections satisfying some partition of unity relations. Equivalently, they are the finite colimits of finite-dimensional commutative C*-algebras. After introducing hypergraph C*-algebras, I will explain the equivalent colimit characterization, another characterization in terms of nonlocal games, and showcase a number of examples. In the second part, I will present a number of undecidability results for hypergraph C*-algebras, based on an undecidability result of Slofstra in combinatorial group theory. Time permitting, I will sketch how this leads to results on the independence of ZFC, and also how these results can be interpreted as statements on the complexity of Hilbert space geometry and on the undecidability of the satisfiability problem in quantum logic. All of this is based on arXiv:1808.09220 and arXiv:1607.05870.