Proper and improper forcing.
2nd ed.

*(English)*Zbl 0889.03041
Perspectives in Mathematical Logic. Berlin: Springer. xlvii, 1020 p. (1998).

The author immediately points out in his introduction that even though this book shares part of its title with his 1982 monograph [Proper forcing (Lect. Notes Math. 940) (1982; Zbl 0495.03035)] and is billed as a second edition, “it is a new book” with six new chapters, some deletions, and extensive revisions. The author’s stated aim is to develop a theory of iterated forcing for the continuum and to provide the reader with methods for proving independence results. He states many results in an “axiomatic” manner so that they may be applied more readily. The intent is that the reader familiar with a solid graduate text in basic set theory [e.g., Kunen, Jech, or Just and Weese] will find this book a complete presentation of proper and improper forcing. The author contends that no previous knowledge of forcing is demanded. This may be unrealistic.

Proper forcing is introduced in chapter III, after two introductory chapters: I. “Forcing, basic facts” and II. “Iteration of forcing”. A notion of forcing \(P\) is proper if for every uncountable cardinal \(\lambda\), \(P\) preserves stationarity modulo the filter \(D_{\aleph_0}(\lambda)\). In III it is shown (amongst other things) that properness is preserved under countable support iteration. Here is a list of the subsequent chapter titles: IV. “On oracle-c.c., the lifting problem of the measure algebra, and ‘\(P(\omega)/\)finite has no non-trivial automorphism’ ”; V. “\(\alpha\)-properness and not adding reals”; VI. “Preservation of additional properties, and applications”; VII. “Axioms and their application”; VIII. “\(\kappa\)-pic and not adding reals”; IX. “Souslin hypothesis does not imply ‘Every Aronszajn tree is special’ ”; X. “On semi-proper forcing”; XI. “Changing cofinalities, equi-consistency results”; XII. “Improper forcing”; XIII. “Large ideals on \(\omega_1\)”; XIV. “Iterated forcing with uncountable support”; XV. “A more general iterable condition ensuring \(\aleph_1\) is not collapsed”; XVI. “Large ideals on \(\aleph_1\) from smaller cardinals”; XVII. “Forcing axioms”; XVIII. “More on proper forcing”. There is an appendix called: “On weak diamond and the power of Ext”. The volume ends with an extensive list of references.

In his introduction the author offers a list of the book’s contents by thirteen subject categories and a second annotated table of contents. This latter annotated listing is extremely important since unfortunately this 1000+ page book has no index. Reviewer’s Comment: I expect the serious user of this important volume will become quite familiar with this second listing.

Proper forcing is introduced in chapter III, after two introductory chapters: I. “Forcing, basic facts” and II. “Iteration of forcing”. A notion of forcing \(P\) is proper if for every uncountable cardinal \(\lambda\), \(P\) preserves stationarity modulo the filter \(D_{\aleph_0}(\lambda)\). In III it is shown (amongst other things) that properness is preserved under countable support iteration. Here is a list of the subsequent chapter titles: IV. “On oracle-c.c., the lifting problem of the measure algebra, and ‘\(P(\omega)/\)finite has no non-trivial automorphism’ ”; V. “\(\alpha\)-properness and not adding reals”; VI. “Preservation of additional properties, and applications”; VII. “Axioms and their application”; VIII. “\(\kappa\)-pic and not adding reals”; IX. “Souslin hypothesis does not imply ‘Every Aronszajn tree is special’ ”; X. “On semi-proper forcing”; XI. “Changing cofinalities, equi-consistency results”; XII. “Improper forcing”; XIII. “Large ideals on \(\omega_1\)”; XIV. “Iterated forcing with uncountable support”; XV. “A more general iterable condition ensuring \(\aleph_1\) is not collapsed”; XVI. “Large ideals on \(\aleph_1\) from smaller cardinals”; XVII. “Forcing axioms”; XVIII. “More on proper forcing”. There is an appendix called: “On weak diamond and the power of Ext”. The volume ends with an extensive list of references.

In his introduction the author offers a list of the book’s contents by thirteen subject categories and a second annotated table of contents. This latter annotated listing is extremely important since unfortunately this 1000+ page book has no index. Reviewer’s Comment: I expect the serious user of this important volume will become quite familiar with this second listing.

Reviewer: J.M.Plotkin (East Lansing)

##### MSC:

03E35 | Consistency and independence results |

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

03E05 | Other combinatorial set theory |

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

03E50 | Continuum hypothesis and Martin’s axiom |