This book presents a set theoretical development for the foundations of the theory of atomic, finitely supported sets and structures. It analyzes whether a classical result can be adequately reformulated by replacing a 'non-atomic structure' with an 'atomic, finitely supported structure’. It  also presents many specific properties, such as finiteness, cardinality, connectivity, fixed point, order, and uniformity, of finitely supported atomic structures that do not have non-atomic correspondents.In the framework of finitely supported sets, the authors analyze the consistency of various forms of choice and related results. They introduce and study the notion of 'cardinality' by presenting various order and arithmetic properties. Finitely supported partially ordered sets, chain complete sets, lattices and Galois connections are studied, and new fixed point, calculability and approximation properties are presented. In this framework, the authors study the finitely supported L-fuzzy subsets of a finitely supported set and the finitely supported fuzzy subgroups of a finitely supported group. Several pairwise non-equivalent definitions for the notion of 'infinity' (Dedekind infinity, Mostowski infinity, Kuratowski infinity, Tarski infinity, ascending infinity) are introduced, compared and studied in the new framework. They provide relevant examples of sets that satisfy some forms of infinity while not satisfying others. Uniformly supported sets are analyzed, and some surprising properties are presented. Finally, some variations of the finite support requirement are discussed.The book will be of value to researchers in mathematical logic and computer science logic.weiterlesen