It is long known that any expansion, M, of the field of real numbers that defines N (the set of all natural numbers) also defines every real Borel set, hence also every real projective set (in the sense of descriptive set theory). Thus, one can easily ask questions about the definable sets of M that turn out to be independent of ZFC (e.g., whether every definable set is Lebesgue measurable). This leads naturally to wondering what can be said about its definable sets if M does not define N. Philipp Hieronymi (Urbana-Champaign) and I have recently obtained a result that can be stated loosely as: M avoids defining N if and only if all metric dimensions commonly encountered in geometric measure theory, fractal geometry and analysis on metric spaces coincide with topological dimension on all images of closed definable sets under definable continuous maps. I will make this statement precise (assuming essentially no knowledge of model theory or dimension theory), explain its significance, and give some easy (yet striking) corollaries and applications.