In this second edition (for the first edition see [Zbl 1237.03001]), the author makes significant alterations in the chapter organization, which include splitting, combining, and extending previous chapters. He also adds three new chapters: (20) Suslin’s problem; (23) Sacks forcing; (27) How many Ramsey ultrafilters exist?
The text itself manifests the author’s four main themes: axiom of choice and ZF, Ramsey theory, cardinal characteristics, and forcing technique. Altogether there are twenty nine chapters, organized into four parts: Part I, Preliminary; Part II, Topics in combinatorial set theory; Part III, From Martin’s axiom to Cohen forcing; Part IV, Combinatorics of forcing extensions. Each chapter ends with Notes that often add historical information, offer further remarks on the chapter’s contents, and/or credit the sources used by the author. Each chapter has its own bibliography; this occasions a certain amount of redundancy, but provides swift access to the relevant literature. Some chapters also have a final section called Related results. There is a list of symbols, a name index, and a subject index.
Here is a brief summary of the highlights of each part.
Part I sets the scene. The four main themes are introduced and the syntax and semantics of first order logic are presented. The completeness and compactness theorems are stated, along with Gödel’s incompleteness theorem. The axioms of Zermelo-Fraenkel set theory (ZF) are described and the terminology needed for the development of the basic theory of sets is introduced. A brief foray into Peano arithmetic together with the incompleteness theorem shows that any proof of the consistency of ZF will require metamathematical assumptions. The completeness theorem and the cumulative hierarchy are used to describe models of ZF, assuming ZF is consistent. The ZF theory of cardinals and ordinals is developed and Zermelo’s axiom of choice (AC) is stated, yielding the extension ZFC of ZF. Part I ends with the development of cardinal arithmetic in ZFC.
Part II begins with a chapter on Ramsey’s theorem and some corolllaries and generalizations thereof. Cardinal relations in ZF are probed and the cardinals \(2^{\aleph_0}\) and \(\aleph_1\) are featured. Equivalents of AC are presented, along with axioms that are weaker than AC. The prime ideal theorem is an example of the latter. As an application of AC, the author presents Hausdorff’s paradoxical decomposition of the sphere and his own version of Raphael Robinson’s 5 element paradoxical decomposition of the ball. Atoms are added to ZF, obtaining the set theory ZFA. Permutation models of ZFA in which AC fails are described, and permutation models due individually to Fraenkel, Mostowski, and Shelah are presented. Thirteen cardinal invariants of the continuum are defined in Chapter 9. These cardinals are \(\leq\mathfrak c\), the cardinality of the continuum, and they describe a combinatorial or analytical property of the continuum. Succeeding chapters are on the shattering number, happy families, and a dual form of Ramsey’s theorem. This last chapter includes proofs of the Hales-Jewitt theorem and the partition Ramsey theorem.
Part III begins with a chapter on Martin’s axiom, which is then followed by a detailed introduction to (Cohen) forcing and a blue print for using forcing to establish consistency and independence results. Symmetric submodels of generic extensions are used to give models where AC fails, e.g., a model in which the reals cannot be well-ordered and one in which every ulltrafilter over \(\omega\) is principal. The Jech-Sochor embedding theorem is proven, showing how to simulate permutation models using symmetric models. Products and iterations of forcings are introduced and used to prove various inequalites (and equalities) involving cardinal invariants. The final chapter of Part III is on Suslin’s problem.
Part IV introduces several eponymous forcings: Sacks forcing, Silver-like forcing, Miller forcing, and Mathias forcing. Of particular concern is the question of what kind of reals are added when a given forcing notion is iterated. It is shown that the addition of Cohen reals is ruled out by the Laver property. Preservation theorems for proper forcing notions are mostly given without proof. Applications to the possible order relationships among the previously introduced cardinal invariants are featured prominently, Shelah’s product tree forcing is used to attack the question of how many Ramsey ultrafilters exist. Combinatorial properties of sets of partitions of \(\omega\) are investigated. In the final chapter, the machinery developed in this text is used to illuminate the Banach-Kuratowski theorem of classical measure theory.
In the reviewer’s judgement, Halbheisen’s book is an excellent source for the intermediate or advanced student of set theory, However, although the reader is asked from time to time to fill in details, there are no exercises in the usual textbook sense. Because of its wealth of material, it should also serve as an excellent resource for those designing advanced courses or searching for seminar assignments for students,