This is a sorely needed compendium of research in the field of computing with sets and other forms of aggregates, such as hypersets. The book guides the reader on a dazzling trip through axiomatization of fragments of set theory, a variety of decision procedures, and the integration of set theory with universal reasoning frameworks (for example, there is a beautiful presentation of logic programming with sets and integration with non-classical logics).

The book is organized in four main parts, accompanied by a nice introduction and a rich and fairly complete bibliography.

Part 1 provides an introduction (chapter 1) to the material in the book, with a justification of its content and a very useful presentation of interconnections between chapters. Chapter 2 provides background material in mathematical logic, including propositional and predicate calculus (syntax, semantics, and deductive mechanisms), and a presentation of the less popular Map language.

Part 2 is dedicated to the development of the foundations of formal reasoning with sets. Chapter 3 leads the reader through the construction of an axiomatic theory for sets. The construction is incremental and covers fairly complex classes of aggregates (such as hyper-sets and “colored” sets). Chapter 4 continues where chapter 3 left off, focusing on the issue of semantics; it develops interpretations of the axiomatic theories, and focuses on identifying models that present desirable computable properties (including some small SETL programs to illustrate the discussion).

Chapter 5 is a little pearl on its own, collecting a variety of examples of uses of set theory in actual problem modeling. Particularly exciting is Section 5.6, dedicated to the use of sets for program verification. Chapter 6 is one of the core chapters of the book. It deals with decision problems in set theory, including limiting results (for example, a set-theoretic encoding of Hilbert’s tenth problem) and a taste of various solvable cases (such as Boolean unification). Chapter 7 covers the topic of inference frameworks appropriate for set theoretic reasoning. Particular attention is given to the description of T-resolution (a resolution framework parameterized by a generic theory T), semantic tableaux, and logic programming.

Part 3 focuses on some more specific decision problems and their solutions. Particularly interesting are chapter 8, which deals with some general aspects of unification in presence of sets, and chapter 10, which provides a stratified view of sets and implications for solving decision problems.

Finally, Part 4 completes the discussion by presenting some case studies of inference frameworks that are appropriate to set theory, such as modal logic (chapter 12) and logic programming (chapter 13).

The book provides an overview of the main issues in the field of computable set theory along with a tutorial presentation of the main lines of research explored. A significant proportion of the techniques and solutions presented are the result of many years of research conducted by the authors, who are recognized experts in the field. The book is organized in a self-contained tutorial format, and is appropriate for use as a textbook for an advanced graduate-level course. Each chapter is enriched by exercises, although no reference to solutions or other supporting material is provided. Particularly useful is the introductory section, which nicely describes the links between the content of the various chapters. This can be used by a teacher to design the syllabus for a course on this topic.

The presentation is rigorous, but not tedious; it is enriched by a variety of beautiful examples and enticing quotations, and is expressed in unusually elegant English. The presentation is sufficiently deep and detailed to be useful for researchers interested in learning about the state of the art in this field.

To my knowledge, this book is unique in presenting issues in computable set theory in a pedagogical format, with a focus on its use in supporting new models of declarative programming systems. The book is a nice companion to Cantone et al. [1], although substantially more accessible and appropriate for a graduate-level course. I highly recommended the book.