The Constituents of Sets, Numbers, and Other Mathematical Objects, Part One

The sets used to construct other mathematical objects are pure sets, which means that all of their elements are sets, which are themselves pure. One set may therefore be within another, not as an element, but as an element of an element, or even deeper, inside several layers of sets within sets. The introduction of the term constituent to describe a set which is within a given set, however deep, induces an apparently novel partial order on sets, and assigns to any given set a diagram which specifies a directed graph, or category, herein dubbed its constituent structure, indicating which sets within it are constituents of which others. Sets with different numbers of elements can have exactly the same constituent structure. Consequently, constituent structure isomorphisms between sets need not preserve the number of elements, although they are still injective, surjective, and invertible. We consider in detail an example of an isomorphism between a one-element set and a five-element set, which is a surjective mapping despite the mismatch in cardinalities. The constituent structure of a set determines the mathematical objects for which the set is a suitable representation. Different schemes for constructing the natural numbers, such as those of von Neumann and Zermelo, generate sets with the same constituent structures. Objects share the constituent structures, not the elements, of the sets used to construct or represent them. The requirement that an object's properties be faithfully encoded within a set's constituent structure and not its non-constituent characteristics such as its cardinality, when made explicit, dictates a specific and novel way of representing ordered pairs and tuples of sets as sets, providing simple formulae for addressing and extracting sets located deep within nested tuples.