For an infinite subset S of the algebraic numbers and a function f on S, mapping to the non-negative real numbers, the Northcott number of S w.r.t f is (if it exists) the minimal t such that the subset of S that maps to real numbers smaller t+\epsilon is infinite for all positive \epsilon. We are interested in the case when f is the house or a (suitably normalised) Weil height, and S is the ring of integers of a field. What are the possible Northcott numbers, and how is this notion connected to decidability questions, going back to Julia Robinson? We will answer these questions. This is joint work with Fabien Pazuki and Niclas Technau.