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Handbook of set theory. In 3 volumes. (English) Zbl 1197.03001
Dordrecht: Springer (ISBN 978-1-4020-4843-2/hbk; 978-1-4020-5764-9/ebook). xiv, pp. 1-736/Vol. 1; xiv, pp. 737-1447/Vol. 2; xiv, pp. 1449-2197/Vol. 3. (2010).

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This Handbook is written for graduate students and researchers, so it is assumed that the reader has already a good basic knowledge in set theory. The 24 chapters and a long introduction are written by acknowledged experts, major research figures in their areas. The Handbook is divided into three volumes where the first one is devoted to Combinatorics, the Continuum, and Constructibility, the second to Elementary Embeddings and Singular Cardinal Combinatorics, and the third one to Inner Models and Determinacy.
In the Introduction (pp. 1–92), Akihiro Kanamori provides a historical and organizational frame for both modern set theory and the Handbook. He recapitulates the consequential historical developments that led to modern set theory as a field of mathematics. He starts with the work of Cantor, Zermelo, Gödel and Cohen, and describes the most important ideas and techniques that had influenced the development of modern set theory, such as infinite combinatorics, the method of forcing, \(0^{\sharp}\), large cardinals, determinacy, Silver’s theorem and covering. He deals with the new expansions that began in the 1980s, for instance the inner model theory of large cardinals, the consistency of determinacy, the work of Shelah on cardinal arithmetic, and Todorcevic’s contributions to combinatorics and topology.
In the following we review the individual chapters of the Handbook.
1. Stationary sets. In this chapter (pp. 93–128), Thomas Jech surveys the work directly involving stationary sets and provides the basic theory of stationary subsets of a regular uncountable cardinal. He describes the possibilities for stationary set reflection and surveys the possibilities of saturation and precipitiousness of the non-stationary ideal. He investigates the closed unbounded filter on \(\mathcal{P}_{\kappa} \lambda\) and deals with applications of the filter \(\mathcal{P}_{\omega_1} A\) in the theory of Boolean algebras. He shows that reflection plays a significant role in applications of Martin’s Maximum and gives a brief description of stationary tower forcing.
2. Partition relations. András Hajnal and Jean A. Larson start this chapter (pp. 129–213) with the classical Ramsey and Erdős-Rado Theorems. They develop a method of Baumgartner, Hajnal and Todorcevic and establish their generalization of the Erdős-Rado Theorem. This method involves taking chains of elementary substructures of the form \(\langle H(\lambda), \in, <^{*},\dots \rangle\). Several of Shelah’s important contributions to the theory are presented, among them a recent result involving strongly compact cardinals and another invoking his pcf theory. The authors investigate partition relations for small countable ordinals. They present the results of Darby and Schipperus which generalize Chang’s result that \(\omega^{\omega} \longrightarrow (\omega^{\omega},3)^2\) is true (with the ordinal exponentiation).
3. Coherent sequences. Stevo Todorcevic gives (on pp. 215–296) a systematic account of his insistent analysis of uncountable order structures, with \(\omega_1\) being both a particular and a paradigmatic case. For \(\omega_1\) the analysis begins with \(C\)-sequences \(\langle C_{\alpha} \mid \alpha < \omega_1 \rangle\), where \(C_{\alpha+1} = \{ \alpha \}\) and, for limit \(\alpha\), \(C_{\alpha}\) is a subset of \(\alpha\) of order type \(\omega\). This gives a ladder system where one can climb up or walk down. The author has shown that walks have a great deal of structure with respect to various distance functions. Systematic versions of special Aronszajn trees are presented, as well as Shelah’s result that adding a Cohen real adds a Suslin tree. He gives applications to Hausdorff gaps, Banach spaces, model theory, graph theory and partition relations. Later on, he deals with a general cardinal \(\kappa\). A range of applications is provided, involving the principle \(\square_{\kappa}\), higher Kurepa trees, and Jensen matrices.
4. Borel equivalence relations. Greg Hjorth provides (on pp. 297–332) a survey of Borel equivalence relations on Polish spaces. For Polish spaces \(X\) and \(Y\), a function \(f : X \rightarrow Y\) is Borel if the preimage of any Borel set is Borel. An equivalence relation on \(X\) is Borel if it is Borel as a subset of \(X \times X\). There is a natural order \(<_B\) on the Borel equivalence relations. The author starts with the seminal Harrington-Kechris-Louveau “Glimm-Effros dichotomy” and discusses various structure theorems, concluding with his work on turbulence. He surveys results on countable equivalence relations and deals with the classification problem.
5. Proper forcing. In this chapter (pp. 333–394), Uri Abraham provides a lucid exposition of Shelah’s proper forcing. He starts with the basic forcing notions and motivates proper forcing. He shows that properness is preserved in countable support iterations and that under CH a length \(\leq \omega_2\) iteration of \(\aleph_1\)-size proper forcings satisfies the \(\aleph_2\)-chain condition. He presents the preservation of \({}^{\omega}\omega\)-bounding properness in countable support iterations. With this, Shelah showed that it is consistent that there are two countable elementarily equivalent structures having no isomorphic ultrapowers by any ultrafilter over \(\omega\). The author shows the preservation of weakly \({}^{\omega}\omega\)-bounding properness. With this, Shelah showed that it is consistent with \(2^{\aleph_0} = \aleph_2\) that the bounding number \(\mathfrak{b}\) is less than the splitting number \(\mathfrak{s}\).
6. Combinatorial cardinal characteristics of the continuum. Andreas Blass investigates (on pp. 395–489) the combinatorial cardinal characteristics of the continuum. He starts with the dominating number \(\mathfrak{d}\), the bounding number \(\mathfrak{b}\), and the splitting number \(\mathfrak{s}\). He describes the Galois-Tukey connections and duality. He introduces several generalizing characteristics corresponding to a given ideal \(\mathfrak{I}\): \({\mathbf{add}} (\mathfrak{I}), {\mathbf{cov}}(\mathfrak{I}), {\mathbf{non}}(\mathfrak{I})\) and cof(\(\mathfrak{I})\). These numbers are investigated for the ideals of meager sets and null sets. He discusses maximal almost disjoint families and independent families. He develops his principle of Near Coherence of Filters, a principle proved consistent by Shelah. He surveys what happens to the characteristics when one iteratively adjoins many generic reals of one kind, dealing with the following reals: Cohen, random, Sacks, Hechler, Laver, Mathias, and Miller.
7. Invariants of measure and category. Tomek Bartoszynski presents (on pp. 491–555) the recent work on measure and category with respect to cardinal invariants. He uses various criteria corresponding to the ideal invariants to define different classes of sets. He develops Borel morphisms that lead to inclusion relations among these classes. Combinatorial characterizations of membership in these classes are given. The author establishes Shelah’s result that it is consistent that \(\text{cf(\textbf{cov}}(\mathfrak{L})) = \omega\). He provides a systematic way of associating to each of the invariants in Cichoń’s diagram a generic real so that iteration with countable support increases that invariant but none of the others.
8. Constructibility and class forcing. Sy D. Friedman provides (on pp. 557–604) the basic theory and applications of class forcing. Class forcing does not in general preserve ZFC, so the first-order property of tameness is isolated, which is necessary and sufficient for this preservation. The most important technique in the subject is the technique of Jensen coding. The author provides a proof of Jensen’s Coding Theorem assuming that \(0^{\sharp}\) does not exist. He gives wide-ranging applications and a nice list of open problems.
9. Fine structure. Ralf Schindler and Martin Zeman provide (on pp. 605–656) an incisive, self-contained account of Jensen’s original fine structure theory. This theory is an in-depth study of definability over levels of constructible hierarchies. It was invented by Jensen and further developed by Jensen, Mitchell, Steel, and others. Fine structure theory is unavoidable for the construction of core models. The authors discuss the pure part of fine structure theory, that is the part which is not linked to any particular kind of constructible models. They use the Mitchell-Steel \(r\Sigma_n\) formulas for discussing iterated projecta and embeddings. The authors analyze fine ultrapowers. They provide a proof, in the absence of \(0^{\sharp}\), of the countably closed weak covering property for \(L\) and a proof of \(\square_{\kappa}\) for \(\kappa > \omega\).
10. \(\Sigma^*\) fine structure. Philip D. Welch presents (on pp. 657–736) the \(\Sigma^*\) fine structure. He starts with the \(\Sigma^*\) hierarchy of formulas and \(\Sigma^*\) ultrapowers and develops a theory of pseudo ultrapowers. These pseudo ultrapowers are used in many combinatorial constructions. He investigates \(\square\) principles and derives a global \(\square\) sequence in \(L\). He discusses variants and generalizations of \(\square\) in the fine-structural inner models. It was shown by Jensen that \(V=L\) implies a global class version of \(\square\). Here, the author gives a \(\Sigma^*\) pseudo-ultrapower proof of this result. At the end of the chapter, the author gives a survey of the extensive work on morasses.
11. Elementary embeddings and algebra. Patrick Dehornoy investigates (on pp. 737–774) the algebraic features of very strong elementary embeddings. Kunen showed that a special strong large-cardinal postulation is inconsistent. So it was natural to investigate weaker postulations. One was that there exists a non-identity elementary embedding \(j : V_{\lambda} \rightarrow V_{\lambda}\) for some limit \(\lambda\). It was shown by Martin that if there is an iterable elementary \(j : V_{\lambda} \rightarrow V_{\lambda}\), then \({\boldsymbol \Pi}^1_2\)-Determinacy holds. Let \(\mathcal{E}_{\lambda}\) be the set of all such elementary embeddings. \(\mathcal{E}_{\lambda}\) is closed under composition \(\circ\) and application of \([\;]\), which is defined by \(j[k] = \bigcup_{j<\lambda} j(k \cap V_{\lambda})\). These operations satisfy the left distributivity law \(j[k[l]] = j[k][j[l]]\). Laver initiated a systematic study of \(\mathcal{E}_{\lambda}\). He established the freeness of the subalgebra generated by one \(j\) subject to the left distributive law. With this analysis, Laver showed that the corresponding word problem for the left distributive law is solvable (with respect to one generator). Dehornoy eliminated the large-cardinal assumption which led to unexpected results about the Artin braid group.
12. Iterated forcing and elementary embeddings. James Cummings gives (on pp. 775–883) a survey of that area of set theory in which iterated forcing interacts with elementary embeddings. He starts with the basic facts about elementary embeddings, and describes how elementary embeddings can be approximated by ultrapowers and by extenders, a special kind of limit ultrapowers. He presents the basics of forcing and iterated forcing, describes how to build generic objects over sufficiently closed inner models, and discusses generic elementary embeddings. He presents a lot of applications. To mention only two, he establishes Magidor’s result that it is consistent that the least strong cardinal is the least measurable cardinal, as well as Baumgartner’s result: If there is a supercompact cardinal \(\kappa\), then there is a forcing extension in which \(\kappa = \omega_2\) and PFA holds.
13. Ideals and generic elementary embeddings. This chapter (pp. 885–1147) by Matthew Foreman covers the techniques of generic elementary embeddings. The difference to classical elementary embeddings is that they are definable in a generic extension of \(V\) rather than in \(V\) itself. The critical points can be as small as \(\omega_1\). The strength of an elementary embedding \(j : V \rightarrow M\) can be described by three parameters: how \(j\) moves the ordinals; how large and closed \(M\) is; the nature of forcing that provided \(j\). The forcing is determined by ideals: One starts with an ideal \(I\) over a cardinal \(\kappa\) and forces with \(\mathcal{P}(\kappa) \setminus I\), then produces an ultrafilter over the ground model \(\mathcal{P}(\kappa)\), and finally gets a generic elementary embedding of the ground model into the corresponding ultrapower. The author provides a wealth of methods and results. We only mention a few: The Shelah-Gitik result that if \(\kappa\) is regular and \(\delta^{+} < \kappa\) then the ideal generated by \(\text{NS}_{\kappa}\) and \(\{ \alpha < \kappa \mid \text{cf}(\alpha) = \delta \}\) is not \(\kappa^{+}\)-saturated; Kunen’s technique for getting an \(\aleph_1\)-complete \(\aleph_2\)-saturated ideal over \(\omega_1\) from a huge cardinal; Foreman’s iteration to get \(\kappa^+\)-complete \(\kappa^+\)-saturated ideals over \(\kappa\) for every regular \(\kappa > \omega\); Woodin’s \(\aleph_1\)-complete \(\aleph_1\)-dense ideal over \(\omega_1\) from an almost huge cardinal.
14. Cardinal arithmetic. Uri Abraham and Menachem Magidor provide (on pp. 1149–1227) a broad-based account of Shelah’s pcf theory and its applications to cardinal arithmetic. They start with a general theory of cardinal functions modulo ideals and cofinal sequences thereof. The major definition in pcf theory is the set \(\text{pcf}(A)\) of possible cofinalities defined for every set \(A\) of regular cardinals, as the collection of all cofinalities of ultraproducts \(\prod A/D\) with ultrafilter \(D\) over \(A\). While the power set can be easily changed by forcing, it is very hard to change \(\text{pcf}(A)\). The authors investigate exact upper bounds and deal with club guessing sequences. They study the ideal \(J_{<\lambda}[A]\) and show that \(J_{<\lambda^+}[A]\) is generated by \(J_{<\lambda}[A]\) together with a single set \(B_{\lambda} \subseteq A\). They establish Shelah’s result that \(2^{\aleph_{\omega}} < \aleph_{\omega_4}\) when \(\aleph_{\omega}\) is strong limit and they present Shelah’s revised GCH result.
15. Successors of singular cardinals. Todd Eisworth provides (on pp. 1229–1350) a well-organized account of the modern theory of successors of singular cardinals. The investigation of combinatorial properties at successors of singular cardinals has emerged as a distinctive subject in modern set theory. Let \(\text{Refl}(\kappa)\) be the assertion that every stationary \(S \subseteq \kappa\) reflects, i.e., there is an \(\alpha < \kappa\) such that \(S \cap \kappa\) is stationary in \(\alpha\). The author discusses how \(\square_{\kappa}\) denies \(\text{Refl}(\kappa^{+})\) and establishes Magidor’s result: If there are infinitely many supercompact cardinals then, in a forcing extension in which they become the \(\aleph_n\)s, \(\text{Refl}(\aleph_{\omega+1})\) holds. He investigates in detail the ideal \(I[\kappa]\) and provides an extensive exploration of scales and weak square principles. He discusses square-brackets partition relations and discusses the existence of Jónsson algebras for successors of singular cardinals.
16. Prikry-type forcing. Moto Gitik presents (on pp. 1351–1447) the Prikry forcing and its applications. The Singular Cardinals Problem is the problem of finding a complete set of rules describing the behavior of the function \(\kappa \rightarrow 2^{\kappa}\) for singular \(\kappa\)s. There are three main tools for dealing with the problem: pcf theory, inner model theory and forcing involving large cardinals. In this chapter the author presents the main forcing tools for dealing with powers of singular cardinals. In the first half, the author deals with countable cofinality. He starts with the basic Prikry forcing. Then he presents the Gitik-Magidor extender-based forcing for adjoining many Prikry sequences with optimal hypothesis. In the second half, the author deals with uncountable cofinality. He presents Radin forcing. He investigates iterations of general Prikry-type forcings and gives a simplified proof of Magidor’s result: It is consistent that the least strongly compact cardinal is the least measurable cardinal.
17. Beginning inner model theory. This chapter (pp. 1449–1495) by William J. Mitchell is the first of several chapters on inner model theory. The author sets out the theory from \(L[U]\) and \(K^{DJ}\) through to inner models of strong cardinals, the coarse theory not requiring fine structure. He presents iteration, comparison, coherence, and coiteration. He considers \(0^{\sharp}\) and other sharps in general, and introduces mice and coherent sequences of (non-overlapping) extenders. He discusses the further developments that involve fine structure.
18. The covering lemma. Here, William J. Mitchell presents (on pp. 1497–1594) variants of the covering lemma and its applications. Jensen’s discovery of the covering lemma arose out of work on the singular cardinals problem. The covering lemma for \(L\) is as follows: If \(0^{\sharp}\) does not exist then for any set \(x\) of ordinals there is a set \(y \in L\) such that \(y \supseteq x\) and \(|y| = |x| + \aleph_1\). The Jensen argument for the covering lemma for \(L\) stimulated the formulation of new inner models. The author discusses variants of the covering lemma and its applications. He outlines a proof of the Jensen and the Dodd-Jensen covering results for \(L\) and \(L[U]\). Further on, the author presents a proof of covering for Mitchell’s core model \(K[\mathcal U]\) for a coherent sequence \(\mathcal U\) of ultrafilters.
19. An outline of inner model theory. John R. Steel presents (on pp. 1595–1684) basic inner model theory in the greatest generality that is currently known. Thus the theory presented here can be applied to models that may satisfy large-cardinal hypotheses as strong as “There is a Woodin cardinal which is a limit of Woodin cardinals”. He starts with the basics of extenders and goes on with iteration trees. They are central for handling overlapping extenders. He deals with the Dodd-Jensen Lemma, solidity and condensation. He provides the \(K^C\) construction and gives applications in descriptive set theory. So, it is shown that the model \(\text{HOD}^{L(\mathbb{R})}\) of all sets hereditarily ordinal-definable in \(L(\mathbb{R})\) is (essentially) an extender model.
20. A core model tool box and guide. Ernest Schimmerling deals here (on pp. 1685–1751) with core model theory at a level where it involves iteration trees. He starts with the basic theory of \(K\) and discusses combinatorial applications of it across set theory. For all but the last section he assumes the anti-large-cardinal hypothesis: There is no proper class model with a Woodin cardinal. He provides a tool section. Here he lists properties of \(K\) that are useful in applications. They include covering properties, absoluteness, complexity and correctness, embeddings of \(K\), and combinatorial principles. In his final section he presents applications of \(K\).
21. Structural consequences of AD. Steve Jackson gives (on pp. 1753–1876) a survey of recent advances in descriptive set theory. He works in the base theory ZF + DC. He starts with a survey of basic notions, such as scales and periodicity, the Coding Lemma, projective ordinals, and Wadge reducibility. In the next section he develops the AD theory of the Suslin cardinals, culminating in a classification theorem. He goes on with a theory of “trivial” description and gives a new proof of the Kechris-Martin Theorem for \(\boldsymbol{\Pi}_3^1\). He provides an introduction to the modern theory of projective sets.
22. Determinacy in \(L(\mathbb{R})\). Itay Neeman investigates (on pp. 1877–1950) the structure \(L(\mathbb{R})\). Woodin showed that AD is equiconsistent with the existence of infinitely many Woodin cardinals. Here the author gives a complete proof of this result. He starts with the Martin-Steel theory of iteration trees. He introduces homogeneous Souslin sets and presents a proof of determinacy for \(\boldsymbol{\Pi}_1^1\) sets from a measurable cardinal. Then he gives a proof of the projective determinacy from Woodin cardinals. This result is improved by reducing the large-cardinal assumption needed for the determinacy of universally Baire sets. He shows that models with Woodin cardinals can be iterated and derives AD in \(L({\mathbb{R}})\).
23. Large cardinals from determinacy. Peter Koellner and W. Hugh Woodin give here an account (on pp. 1951–2119) of Woodin’s technique for deriving large-cardinal strength from determinacy hypotheses. These results appear here for the first time. The heart of the chapter is the Generation Theorem that provides a template for generating Woodin cardinals from refined determinacy hypotheses. They start with a proof of Solovay’s result that \(\omega_1\) is measurable under ZF + AD. The heigths are reached when they show that (in a certain strong determinacy context) HOD can contain many Woodin cardinals, and the central Generation Theorem. They introduce the notion of strategic determinacy. With this, they establish the Generation Theorem and a number of instantial cases. They show that the Generation Theorem can be iteratively applied to generate infinitely many Woodin cardinals. They describe a reduction to second-order Peano Arithmetic.
24. Forcing over models of determinacy. Paul B. Larson describes (on pp. 2121–2177) the work of Woodin on forcing over models of determinacy. It was shown by Woodin: If there exists a measurable cardinal which is greater than infinitely many Woodin cardinals, then the Axiom of Determinacy holds in \(L(\mathbb{R})\). The following was also shown by Woodin: If \(\delta\) is a limit of Woodin cardinals and there exists a measurable cardinal greater than \(\delta\), then no forcing construction in \(V_{\delta}\) can change the theory on \(L(\mathbb{R})\). Central in the work of Woodin is the partial order \({\mathbb{P}}_{\text{max}}\). This partial order produces an extension of \(L({\mathbb{R}})\) whose \(H(\omega_2)\) is the direct limit of the structures \(H(\omega_2)\) of models satisfying every forceable theory. Here, the author gives a complete account of the basic analysis of the \(\mathbb{P}_{\text{max}}\) extension of \(L(\mathbb{R})\), relative to published results. Then he investigates variations of \(\mathbb{P}_{\text{max}}\) and forcing over larger models of determinacy. He also briefly introduces Woodin’s \(\Omega\)-logic.
The Handbook is completed by an extensive Index (pp. 2179–2197).

03-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to mathematical logic and foundations
03Exx Set theory
00B15 Collections of articles of miscellaneous specific interest
03E02 Partition relations
03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E10 Ordinal and cardinal numbers
03E15 Descriptive set theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models
03E45 Inner models, including constructibility, ordinal definability, and core models
03E55 Large cardinals
03E60 Determinacy principles
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