Transfinite nesting in array-theoretic figures, changes, rigs, and arms. Part I

Abstract

Nesting and stemming (infinite successive singling) of arrays of nestings and stemmings result in forms. Forms of 0th-, 1st-, 2nd-, or 3rd-order, array-theoretic, totally defined functions are again such functions, called, respectively, figures, changes, rigs, and arms. One arms a rig before rigging a change before changing a figure. Part I lays the foundation for a new approach to a theory of arrays. This Part considers the analogy between array-theoretic and Euclidean figures, analyzes form separately from substance, introduces N th-order functions, presents the beginnings of a syntax for the theory, and constructs a formal system to deduce the first few consequences of the first two primitive operations.