 Currently, there seems to be even less consensus on how (and whether) one should define singular structure based on the idea of missing points than there is regarding definitions based on path incompleteness. Moreover, this project also faces even more technical and philosophical problems. For these reasons, path incompleteness is generally considered the default definition of singularities. For the remainder of this article, therefore, singular structure will be assumed to be characterized by incomplete, inextendible paths, with the exception of the discussion of  section 1.4  below. 