Set theory. (Teorie množin). (Czech) Zbl 0635.03039

Balcar, Bohuslav; Štěpánek, Petr

Praha: Academia. 412 p.; Kčs 55.00 (1986).

Books on set theory can be divided into two classes. In one class fall textbooks presenting the classical material known in essence already to Cantor and codified in its final form before the Second World War. In the other class there are monographs on various aspects of the revolutionary developments that have taken place in the meantime (such as Gödel’s constructible universe, Cohen’s forcing, large cardinals, etc.). A unified presentation of those fundamental achievements of both the classical and modern period that are nowadays indispensable to all mathematicians working in general topology, abstract algebra or other set-theoretically grounded disciplines was lacking. The authors of the work under review undertook to write exactly such a book and, on the whole, succeeded admirably. Their emphasis is on mathematics, especially combinatorial principles and their consequences, rather than metamathematics and proofs of consistency/independence. For the most part, they selected just the right topics and the best available proofs. Their style is highly polished and very readable and the number of mistakes (mostly misprints) amazingly small. I would choose this book for a text in a first-year graduate (U.S.) level course in set theory though the book has no exercises and it is in Czech. Neither of these drawbacks is beyond remedy in future editions. For the non-Czech readers I shall outline the contents of the book, together with some further comments, in more detail than is customary in reviews.
The book begins with a brief survey of the highlights in the history of the subject. Then, following a section discussing the language of set theory, the authors state the Zermelo-Fraenkel axioms and discuss procedures for extending the language of ZF by class terms and (unquantified) class variables. Subsequently, classes are consistently used throughout the book. Use of classes makes possible concise statements and proofs of many results and is almost inevitable in any discussion of models of set theory, but it makes the first three sections of Chapter I rather heavy going for readers with no familiarity with formal logic. It might be wise to heed the authors’ advice and only skim through this material at first reading. The remaining sections of Chapter I deal with the basic concepts of classical set theory in ways that are now more or less standard. Topics covered include relations, functions, equivalences and orderings, finite sets and natural numbers. Equipotence and subpotence relations are defined, Cantor-Bernstein and Cantor Theorems proved, and the Continuum Hypothesis stated. There follows a section on the Axiom of Choice and its usual equivalents. AC is used freely in most of the rest of the book. Finally, filters, ideals and measures are introduced and studied, limits of sequences relative to a filter are defined and applied to the construction of Banach limits. Ultrafilters are used to prove the Compactness Principle, whose usefulness is illustrated by proving that an ultrafilter is fixed by a mapping f iff f is an identity almost everywhere, and by other examples. One serious omission in this chapter is the Recursion Theorem, which is never explicitly stated or proved, but is used implicitly in several proofs. (The Transfinite Recursion Theorem is proved in Chapter II but without any mention of recursion on \(\omega\) as an important special case.)
The bulk of Chapter II is concerned with ordinal and cardinal numbers. The usual von Neumann definition of ordinals is employed and ordinal arithmetic is pursued far enough to give a proof of Goodstein Theorem. Cardinal numbers and cardinal arithmetic come next. It is shown how cardinal exponentation is determined by the behavior of \(\aleph_{\alpha}^{cf(\aleph_{\alpha})}\), Silver’s Theorem is stated (and proved in Chapter III) and results of Galvin, Hajnal and Shelah on the behavior of exponentiation at singular cardinals are stated without proof. Singular Cardinal Hypothesis is discussed and the simplification of cardinal arithmetic it yields is derived. The remaining sections of Chapter II could be omitted by many readers. In § 6 the authors study well-founded relations, define transitive closure and rank, prove Mostowski Collapsing Lemma and show that the Axiom of Foundation is equivalent to \(V=WF\). § 7 deals with the constructible universe L, but the treatment is incomplete. The purpose seems to be mainly to provide some background for subsequent references to various consistency results obtained with the help of L. Thus the hierarchy \(L_{\alpha}\) is defined using \(F_ 1,...,F_ 7\), but it is not shown that every definable subset of \(L_{\alpha}\) can be constructed using only these operations. The proof that \(V=L\) implies AC and GCH is outlined, but the order of presentation is confusing (Lemma 7.5 (iv) (\(\alpha\geq \omega \Rightarrow | L_{\alpha}| =| \alpha |)\) requires AC for its proof (this is not pointed out), but the canonical well-orderings \(\triangleleft_{\alpha}\) of \(L_{\alpha}\) are constructed only in Lemma 7.8) and the crucial step \((x\subseteq \omega_{\alpha}\Rightarrow x\in L_{\beta}\) for \(\beta <\omega_{\alpha +1})\) is not proved (although it could be outlined with the same amount of detail as given elsewhere in this section). The authors then discuss the relationship of cardinal numbers and cofinalities in L versus V, state Jensen’s Covering Theorem and prove that it implies the Singular Cardinal Hypothesis.
Chapter III is the best in the book. It is devoted to modern achievements of combinatorial set theory and provides an excellent introduction into its techniques as well as representative results. § 1 deals with constructions of independent systems, almost disjoint systems, \(\Delta\)- systems and free sets. A typical application: the Suslin number of \(^{\alpha}2\) (in the product topology) is \(\aleph_ 0\) (if \(\alpha\geq \omega)\). § 2 studies closed unbounded and stationary sets. Fodor’s Theorem on regressive functions and Solovay’s result on decomposition of stationary sets are proved here, as well as the above-mentioned theorem of Silver. Ulam matrices are introduced. Next follows the statement of Jensen’s principle \(\diamond\) and its variants, Kunen’s proof that \(\diamond_{\lambda}(E)\) is equivalent to \(\diamond '_{\lambda}(E)\), Gregory’s proof that GCH implies \(\diamond_{\omega_ 2}(E(\omega))\), the statement of \(\square_{\lambda}\) and proofs of some of its consequences for reflection of stationary sets. § 3 is titled “Trees and Linear Orderings”. The main topics are: existence of large systems of incomparable types of orderings of sets of reals, trees, esp. a construction of an Aronszajn tree on \(\kappa^+\) if \(\kappa^{<\kappa}=\kappa\) and Jensen’s constructions of Suslin trees (on \(\kappa^+\) assuming \(\kappa^{<\kappa}=\kappa\) and \(\diamond_{\kappa^+}(E(\kappa))\) and on arbitrary \(\lambda >\omega\) assuming a \(\square\)-like principle). The relationship between Aronszajn (resp. Suslin, Kurepa) trees and Specker (resp. Suslin, Kurepa) orderings is studied in detail. Weak Kurepa Hypothesis is used to prove that all uniform ultrafilters on \(\omega_ 1\) are regular. § 4 is devoted to partition properties. Classical Ramsey Theorem and its canonical version come first. The proof of Galvin-Prikry Theorem is next, followed by Laver’s proof of Fraissé Conjecture. The section concludes with the Erdős-Radó partition theorem and some of its topological applications. Throughout this chapter the authors intersperse numerous references to relative consistency of various related results, with frequent mentions of large cardinals. So the reader may wellcome § 5, which provides definitions of the most important kinds of large cardinals and surveys some of their properties and interdependencies, albeit mostly without proof.
The final Chapter IV is devoted to Boolean algebras and generic extensions of models of set theory. Although it is as much a pleasure to read as the rest of the book, it seems to me that a different choice of material (for example, more applications of combinatorics, the ultraproduct construction, much more on large cardinals) would be more in the spirit of the book. The chapter is too brief and, in a way, too polished to give readers good working knowledge of the Boolean-valued models of set theory (for example, forcing is only defined 4 pages before the end of the book, and never used). This is a point where metamathematics becomes an intrinsic part of the subject, and it would be best to leave this material to a second course, especially as some excellent full-length presentations of Boolean-valued models are already available. As it is, § 1 of Chapter IV defines Boolean algebras and complete Boolean algebras and gives some examples (clopen sets, regular open sets, Borel sets etc.). The highlights of § 2 are the Stone representation, the construction of a complete Boolean algebra from any ordering, study of distributivity, Lévy’s collapsing algebra and its characterization due to McAloon. Finally in § 3 the authors define the most crucial metamathematical concepts (relativization, absoluteness and the notion of a transitive model of ZF). Following Vopěnka they say that a set \(\sigma\subseteq a\in M\) is generic over a transitive model M of ZF if \(\{\) r”\(\sigma\) : r is a relation, \(r\in M\}\) is closed under differences, and prove that every generic set can be represented as a generic ultrafilter over some M-complete Boolean algebra. At this point they introduce names for subsets of M and state the Generic Model Theorem. Its proof via the usual Boolean universe concludes the section and the book; in-between, the authors work out four examples: a model where \(2^{\omega}=\omega_ 1\) (constructed by collapsing), a model for \(\diamond\), Cohen’s model for \(2^{\omega}>\omega_ 1\) and Baumgartner’s results showing existence of models where \(2^{\omega}>\omega_ 2\) but every system of almost disjoint sets on \(\omega_ 1\) has cardinality \(\leq \omega_ 2\). In addition, this section also contains the statement of Martin’s Axiom and proofs of some of its consequences.

Reviewer: K.Hrbacek

MSC:

03-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
03E05	Other combinatorial set theory
03E10	Ordinal and cardinal numbers
03E35	Consistency and independence results

Keywords:

ordinal numbers; constructible sets; closed unbounded sets; combinatorial principles; classes; relations; functions; equivalences; orderings; natural numbers; filters; ideals; measures; cardinal numbers; ordinal arithmetic; cardinal arithmetic; combinatorial set theory; independent systems; almost disjoint systems; \(\Delta \)-systems; free sets; stationary sets; trees; partition properties; relative consistency; large cardinals; Boolean algebras; generic extensions of models of set theory; Boolean-valued models; generic ultrafilter