 A path in spacetime is a continuous chain of events through space and time. If I snap my fingers continually, without pause, then the collection of snaps forms a path. The paths used in the most important singularity theorems represent possible trajectories of particles and observers. Such paths are known as world-lines; they consist of the continuous sequence of events instantiated by an object's existence at each instant of its lifetime. That the paths be incomplete and inextendible means, roughly speaking, that, after a finite amount of time, a particle or observer following that path would “run out of world”, as it were—it would hurtle into the tear in the fabric of spacetime and vanish. (See  Figure 1.)  Alternatively, a particle or observer could leap out of the tear to follow such a path. While there is no logical or physical contradiction in any of this, it appears on the face of it physically suspect for an observer or a particle to be allowed to pop in or out of existence right in the middle of spacetime, so to speak—if that does not suffice for concluding that the spacetime is singular, it is difficult to imagine what else would. At the same time as this criterion for singularities was first proposed, the ground-breaking work predicting the existence of such pathological paths (Penrose 1965, 1968; Hawking 1965, 1966a, 1966b, 1966c, 1966d; Geroch 1966, 1967, 1968b, 1970; Hawking and Penrose 1970) produced no consensus on what ought to count as a necessary condition for singular structure according to this criterion, and thus no consensus on a fixed definition for it. 