Splitting the Cartesian point

Abstract

It is argued that the point structure of space and time must be constructed from the primitive “extensional” character of space and time. A procedure for doing this is laid down and applied to one-dimensional and two-dimensional systems of abstract “extensions.” Topological and metrical properties of the constructed point systems, which differ nontrivially from the usual ℝ and ℝ² models, are examined. Briefly, constructed points are associated with “directions” and the Cartesian point is split. In one-dimension each point splits into a point pair compatible with the linear ordering. An application to one-dimensional particle motion is given, with the result that natural topological assumptions force the number of “left point, right point” transitions to remain locally finite in a continuous motion. In general, Cartesian points are seen to correspond to certain filters on a suitable Boolean algebra. Constructed points correspond to ultrafilters. Thus, point construction gives a natural refinement of the Cartesian systems.