At the heart of the (weak and strong) cosmic censorship conjectures is a statement regarding singularity formation in general relativity. Even in spherical symmetry, cosmic censorship seems, at the moment, mathematically intractable. To give a framework in which to address these very difficult problems, we will introduce a notion of spherically symmetric ‘strongly tame’ Einstein-matter models, examples of which are given by Einstein-Maxwell-Klein-Gordon (charged scalar fields) and Einstein-Maxwell-dilaton (solutions of which appear in the low-energy limit of string theory). We will demonstrate that for any ‘strongly tame’ model there is an a priori characterization of the spacetime boundary. In particular, for any ‘strongly tame’ Einstein-matter model, a ‘first singularity’ must emanate from a spacetime boundary to which the area-radius r extends continuously to zero. Proving this result relies on a weak extension (breakdown) criterion involving only bounds on the area-radius and spacetime volume.