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Descriptive set theory and definable forcing. (English) Zbl 1037.03042

Mem. Am. Math. Soc. 793, 140 p. (2004).
The subject of the book is the relationship between definable forcing and descriptive set theory. Most of the results assume some strong large cardinal axioms which can be detected more precisely from proofs. The book is written for specialists of the subject and a large amount of the notation is not introduced directly in the main text but can be found in four appendices at the end of the book. The exposition is far from being elementary. The book is divided into five chapters. Chapter 1 is a short introduction to the subject and the structure of the book. Chapter 2 is a discussion of definable forcing notions adding a single real. Let \(I\) be a \(\sigma\)-ideal on \(\mathbb R\) and let \(P_I=B(\mathbb R)\setminus I\) be the family of Borel sets in \(\mathbb R\) which are not members of \(I\). The set \(P_I\) is ordered by inclusion. The interest for partial orders \(P_I\) is justified by this representation theorem: The forcing extension by the poset \(P_I\) is given by a single real and conversely, every suitably definable partial ordering adding a single real is forcing equivalent to a poset \(P_I\) for a suitable \(\sigma\)-ideal \(I\) (under some large cardinal axiom). The theory on posets \(P_I\) is followed by several examples of ideals \(I\). Namely, there is some discussion on the \(\sigma\)-ideals of countable sets, \(\sigma\)-compact sets, meager sets, null sets, \(\sigma\)-porous sets, sets of finite \(s\)-dimensional Hausdorff measure, \(\sigma\)-ideals connected with Laver forcing, Mathias forcing, Silver forcing, and several other more or less known \(\sigma\)-ideals. The author checks some descriptive set-theoretic properties for these ideals and he looks for “combinatorially manageable” dense subsets of the posets \(P_I\), frequently consisting of closed sets. In Chapter 3 he proves that for iterable ideals \(I\) the countable support iteration \(P_\kappa\) of \(P_I\) is isomorphic to the poset \(P_\kappa\) consisting of all Borel \(I\)-perfect subsets of \(\mathbb R^X\) for all countable sets \(X\subseteq\kappa\). For countable ordinals \(\alpha\), the \(\alpha\)th Fubini power \(I^\alpha\) of the ideal \(I\) defined by means of an infinite game of two players is a \(\sigma\)-ideal on \(\mathbb R^\alpha\). Some of the proved dichotomy results imply that \(P_\alpha\) is a dense subset of \(B(\mathbb R^\alpha)\setminus I^\alpha\) for countable \(\alpha\). Descriptive properties of ideals \(I^\alpha\) are proved and some cardinal invariants are computed. In Chapter 4 the author shows how to iterate certain forcing notions along any linear ordering and investigates the ideals associated with these iterations. The main results of the book are contained in Chapter 5. The Ciesielski-Pawlikowski axiom CPA, which holds in the iterated Sacks model, is considered together with its generalization \(\text{CPA}(I)\), obtained by replacing the ideal of countable sets by any \(\sigma\)-ideal on \(\mathbb R\) and Sacks forcing by \(P_I\). The first of the main results says that the inequality between a tame cardinal invariant and \(\text{cov}I\) can be forced if and only if it can be proved from \(\text{CPA}(I)\). The duality between \(\text{cov}I\) and \(\text{non}I\) fails in general, but \(\text{cov}I=\mathfrak c\) implies \(\text{non}I\leq\aleph_2\) for projective ideals \(I\) under some circumstances. Next, some interpolation theorems on implications from some assertions about tame cardinal invariants to commonly known principles of set theory are proved. Finally, the author proves several preservation theorems for countable support iterations of forcing notions.

MSC:

03E15 Descriptive set theory
03E17 Cardinal characteristics of the continuum
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03E40 Other aspects of forcing and Boolean-valued models
03E55 Large cardinals
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