From the preface: “This book is an essay: not a monograph, or a textbook, but an essay. It is intended to be a good read for those people who are already interested in this topic (or think they might become interested in it) rather than a comprehensive treatment for people who wish to master it, and a reference work for those who already have. ... I have attempted to touch all the major areas of set theory with a universal set, though I admit to not having made any serious attempt with Skala, and I have kept a low profile where positive set theory is mentioned. ... The result is a book that concentrates heavily on NF. ... To a certain extent, this is unavoidable: NF is a much richer and more mysterious system than the other set theories with a universal set, and there are large areas in its study (e.g. the reduction of the consistency question) which have no counterparts elsewhere in the study of set theories with \(V\). There just is a great deal more to say about NF than about the other systems. Although I have not attempted to discuss all open questions, I have tried to cover those that seemed interesting to me, to explain why they seemed interesting, and to try to give pointers for people who wish to pursue topics that I do not. The coverage of Rieger-Bernays permutation models is the most comprehensive to be had, and the treatment of the theories in the tradition of Church universal set theory is likely to remain the most extensive of any in print until Sheridan finishes {his D. Phil. thesis}.”
A summary of the table of contents goes like this: 1. Introduction (1.1 Annotated definitions; 1.2 Some motivations and axioms; 1.3 A brief survey; 1.4 How do theories with \(V\in V\) avoid the paradoxes? 1.5 Chronology). 2. NF and related systems (2.1 NF; 2.2 Cardinal and ordinal arithmetic; 2.3 The Kaye-Specker equiconsistency lemma; 2.4 Remarks on subsystems, term models, and prefix classes; 2.5 The converse consistency problem). 3. Permutation models (3.1 Permutations in NF; 3.2 Applications to other theories). 4. Interpretations in well-founded sets (4.1 Church’s universal set theory CUS; 4.2 Mitchell’s set theory; 4.3 Beyond Church, Sheridan, and Mitchell). 5. Open Problems. 6. Bibliography. (The Bibliography is an extensive eleven page listing of items concerning set theory with a universal set).