The argument of Part I of this book may be summed up as follows. The notion of causality can be defined as a partial order on an infinite set of geometrical points. Defining causality is the same as assigning a light cone at each point (of a space). A light cone is determined by the set of light rays through its vertex. A light ray is totally ordered by the natural past-future order on it. The partial order on the whole space, called the causal order, may be reconstructed – i.e., axiomatized – from the properties of light rays and their intersections. A key property of a light ray is that between any two points of it lies a third, i.e., the cardinality of the set of points constituting a light ray is at least ℵ0. It turns out that the entire causal structure can be defined on a set of cardinality ℵ0. In common parlance (though not in the technical topological sense), such spaces would be called discrete, as opposed to continuua. Analysis of the causal spaces so defined shows that the locally compact ones among them can be densely embedded in continuua that have the local structure of finite-dimensional differentiable manifolds. Thus, although causality does not imply that space-time is a differentiable manifold, it comes as close as possible, mathematically, to implying it without actually doing so; a principle that most people would consider purely physical has far-reaching mathematical implications.