Set theory.

*(English)*Zbl 1262.03001
Studies in Logic (London) 34. London: College Publications (ISBN 978-1-84890-050-9/pbk). viii, 401 p. (2011).

This is an important textbook in advanced set theory, comparable to T. Jech’s [Set theory. The third millenium edition. Berlin: Springer (2003; Zbl 1007.03002)] and to [M. Foreman (ed.) and A. Kanamori (ed.), Handbook of set theory. In 3 volumes. Dordrecht: Springer (2010; Zbl 1197.03001)]. It assumes expert knowledge of elementary axiomatic set theory (for example, from the author’s textbook [Set theory. An introduction to independence proofs. Amsterdam, New York, Oxford: North-Holland Publishing Company (1980; Zbl 0443.03021)]) and model theory. There are five chapters.

Chapter I, labeled “Background material”, starts with a short discussion about the author’s conception of the formalist philosophy of mathematics and introduces the Zermelo-Fraenkel axiom system, followed by a treatment of many of the basic topics (including ordinals, induction, recursion, uncountable cardinals, the axiom of choice, and a few results in model theory and recursion theory).

Chapter II, “Easy consistency proofs”, starts by proving the relative consistency of the Axiom of Foundation, and then derives the Reflection Theorem. This is followed by a nice treatment of constructible sets and \(V = L\), and then by the definition of ordinal-definable sets and a sketch of the relative consistency of ZFC. The chapter concludes with a treatment of set theory with classes.

Chapter III, “Infinitary combinatorics”, has much more meat to it: the countable chain condition, Martin’s Axiom, trees, club filters, the diamond principles, and the theory of elementary submodels.

Chapter IV contains a nice exposition of forcing and some of its applications, including the independence of the axiom of choice.

Chapter V, “Iterated forcing”, has some sophisticated applications, especially to some independence proofs involving Martin’s Axiom. It ends with a relatively modest exposition of the Proper Forcing Axiom. Unfortunately, we have not succeeded in presenting an adequate summary of the rich results that can be found in this book.

Chapter I, labeled “Background material”, starts with a short discussion about the author’s conception of the formalist philosophy of mathematics and introduces the Zermelo-Fraenkel axiom system, followed by a treatment of many of the basic topics (including ordinals, induction, recursion, uncountable cardinals, the axiom of choice, and a few results in model theory and recursion theory).

Chapter II, “Easy consistency proofs”, starts by proving the relative consistency of the Axiom of Foundation, and then derives the Reflection Theorem. This is followed by a nice treatment of constructible sets and \(V = L\), and then by the definition of ordinal-definable sets and a sketch of the relative consistency of ZFC. The chapter concludes with a treatment of set theory with classes.

Chapter III, “Infinitary combinatorics”, has much more meat to it: the countable chain condition, Martin’s Axiom, trees, club filters, the diamond principles, and the theory of elementary submodels.

Chapter IV contains a nice exposition of forcing and some of its applications, including the independence of the axiom of choice.

Chapter V, “Iterated forcing”, has some sophisticated applications, especially to some independence proofs involving Martin’s Axiom. It ends with a relatively modest exposition of the Proper Forcing Axiom. Unfortunately, we have not succeeded in presenting an adequate summary of the rich results that can be found in this book.

Reviewer: Elliott Mendelson (Flushing)

##### MSC:

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

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

03E35 | Consistency and independence results |

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

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

03E50 | Continuum hypothesis and Martin’s axiom |

03E55 | Large cardinals |

03E57 | Generic absoluteness and forcing axioms |