Could Extended Objects Be Made Out of Simple Parts? An Argument for "Atomless Gunk"

Let us say that an object is wholly (or entirely) of simples just in case it is an aggregate of absolutely parts spread throughout an region-that is, just in case there is a set S such that: (a) every member is a point-sized part of the object, and (b) for every x, x is part of the object if and only if it has a part in common with some member of S. Could a truly substance be composed entirely of (simple) parts? Reflection upon the fact that it must be at least possible for objects to touch one another suggests that the answer to this question is: No. (For the purposes of this paper, it will be convenient to use extended throughout to mean three-dimensionally, spatially extended, and unextended to mean dimensionless or having the dimension of a geometrical point.) Although Zeno's mathematical paradoxes of plurality were long thought to raise insurmountable difficulties for the supposition that an thing could be composed of simple parts, Adolf Grunbaum has shown that these paradoxes are significantly defused by Cantor's discovery of the distinction between denumerably and non-denumerably infinite numbers.' If Grtinbaum is right, the traditional reasons for doubting the consistency of conceiving of an continuum as an aggregate of elements have been laid to rest,2 and the question of this paper is ripe for reconsideration. Numerous commonplaces of the modern analytic geometry of physical space will be assumed in the arguments to follow. Perhaps this is a mistake;