Set theory. The third millennium edition, revised and expanded.

*(English)*Zbl 1007.03002
Springer Monographs in Mathematics. Berlin: Springer. xiii, 769 p. (2003).

Jech’s classic monograph [Set theory. Academic Press, New York (1978; Zbl 0419.03028); Set theory. 2nd, corr. ed. Springer, Berlin (1997; Zbl 0882.03045)] has been a standard reference for a generation of set theorists (including the reviewer). Though carrying the same title and labeled “The Third Millenium Edition,” the present work is in fact a new book. The material has been rearranged into three parts, the third of which mostly deals with recent developments of the field which have taken place since the publication of the first edition. Even sections presenting older results have been rewritten and modernized. Exercises have been moved to the end of each section. The bibliography, the section on notation, and the index have been considerably expanded as well. This new edition will certainly become a standard reference on set theory for years to come.

After presenting the Zermelo-Fraenkel axioms of set theory, the first part, called “Basic Set Theory,” provides an introductory exposition of the main notions and areas of set theory, like ordinals and cardinals, real numbers (including basic descriptive set theory), some combinatorial set theory, a glimpse of large cardinals, and basics on models of set theory. This part is built up coherently and may well serve as basis for a one-semester introductory course in set theory.

The main body of the book, “Part II. Advanced Set Theory,” contains the important techniques and ideas of modern set theory, dealing with constructibility, forcing, large cardinals, combinatorial set theory, and descriptive set theory, as well as with the interplay between these areas. Most of the material in this part also appeared in the first edition. Exceptions include an outline of the proof of Jensen’s Covering Theorem (Section 18), a brief discussion of inner models for sequences of measures (Section 19) and of extenders (Section 20), results on precipitousness and saturation of the nonstationary ideal (Section 23), and a brief discussion of cardinal invariants related to measure and category (Section 26). The most notable addition is, perhaps, an introduction to Shelah’s pcf theory in Section 24, which contains a complete proof of his result saying that if \(\aleph_\omega\) is a strong limit cardinal then \(2^{\aleph_\omega} < \aleph_{\omega_4}\). Apart from a few sketchy proofs, details in Part II are carefully worked out, and this is the place to study set-theoretic techniques in depth.

Part III, entitled “Selected Topics,” presents an overview of current directions of research in set theory from areas as diverse as inner model theory (e.g., fine structure in Section 27 and core models in Section 35), forcing theory (e.g., the open coloring axiom in Section 29, proper forcing and PFA in Section 31, and semiproper forcing and Martin’s Maximum in Section 37), descriptive set theory (e.g., determinacy in Section 33), as well as the connection of these areas with large cardinals. In particular, a number of equiconsistency results are mentioned (e.g., consistency strength of AD in Section 33, of the failure of the singular cardinal hypothesis in Section 36, of density and saturation of the nonstationary ideal in Section 38). Proofs are often sketchy and technical details are mostly missing. This reflects the fact that, unlike at the time of the first edition, a comprehensive treatment of all major techniques of set theory in a single volume is no longer possible. Because of the breadth of areas covered, the third part is a good place to get an idea on what current research in set theory is about and to look up recent developments.

After presenting the Zermelo-Fraenkel axioms of set theory, the first part, called “Basic Set Theory,” provides an introductory exposition of the main notions and areas of set theory, like ordinals and cardinals, real numbers (including basic descriptive set theory), some combinatorial set theory, a glimpse of large cardinals, and basics on models of set theory. This part is built up coherently and may well serve as basis for a one-semester introductory course in set theory.

The main body of the book, “Part II. Advanced Set Theory,” contains the important techniques and ideas of modern set theory, dealing with constructibility, forcing, large cardinals, combinatorial set theory, and descriptive set theory, as well as with the interplay between these areas. Most of the material in this part also appeared in the first edition. Exceptions include an outline of the proof of Jensen’s Covering Theorem (Section 18), a brief discussion of inner models for sequences of measures (Section 19) and of extenders (Section 20), results on precipitousness and saturation of the nonstationary ideal (Section 23), and a brief discussion of cardinal invariants related to measure and category (Section 26). The most notable addition is, perhaps, an introduction to Shelah’s pcf theory in Section 24, which contains a complete proof of his result saying that if \(\aleph_\omega\) is a strong limit cardinal then \(2^{\aleph_\omega} < \aleph_{\omega_4}\). Apart from a few sketchy proofs, details in Part II are carefully worked out, and this is the place to study set-theoretic techniques in depth.

Part III, entitled “Selected Topics,” presents an overview of current directions of research in set theory from areas as diverse as inner model theory (e.g., fine structure in Section 27 and core models in Section 35), forcing theory (e.g., the open coloring axiom in Section 29, proper forcing and PFA in Section 31, and semiproper forcing and Martin’s Maximum in Section 37), descriptive set theory (e.g., determinacy in Section 33), as well as the connection of these areas with large cardinals. In particular, a number of equiconsistency results are mentioned (e.g., consistency strength of AD in Section 33, of the failure of the singular cardinal hypothesis in Section 36, of density and saturation of the nonstationary ideal in Section 38). Proofs are often sketchy and technical details are mostly missing. This reflects the fact that, unlike at the time of the first edition, a comprehensive treatment of all major techniques of set theory in a single volume is no longer possible. Because of the breadth of areas covered, the third part is a good place to get an idea on what current research in set theory is about and to look up recent developments.

Reviewer: Jörg D.Brendle (Kobe)

##### MSC:

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03Exx | Set theory |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

03E05 | Other combinatorial set theory |

03E45 | Inner models, including constructibility, ordinal definability, and core models |

03E15 | Descriptive set theory |

03E40 | Other aspects of forcing and Boolean-valued models |

03E50 | Continuum hypothesis and Martin’s axiom |

03E55 | Large cardinals |

03E35 | Consistency and independence results |

03E60 | Determinacy principles |

03E04 | Ordered sets and their cofinalities; pcf theory |

03E10 | Ordinal and cardinal numbers |

03E17 | Cardinal characteristics of the continuum |

03E02 | Partition relations |