In this paper, we present an axiomatic formulation of Discrete Random Sets, and extend Choquet's uniqueness result to obtain a recursive procedure for the computation of the underlying event- space probability law, given a consistent Discrete Random Set specification via its generating functional. Based on this extension, we investigate the structure of Discrete Random Set models that enjoy the properties of independent decomposition/superposition, and present a design methodology for deriving models that are guaranteed to be consistent with some underlying event-space probability law. These results pave the way for the construction of various interesting models, and the solution of statistical inference problems for Discrete Random Sets.