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The covering property axiom, CPA. A combinatorial core of the iterated perfect set model. (English) Zbl 1066.03047

Cambridge Tracts in Mathematics 164. Cambridge: Cambridge University Press (ISBN 0-521-83920-3/hbk; 0-511-20845-6/e-book). xxi, 174 p. £ 40.00; $ 70.00 (2004).
The Covering Property Axiom (CPA) arose in connection with the investigation of the iterated Sacks model, that is, the iterated perfect set model. Many interesting mathematical properties, especially those concerning real analysis, are known to be true in the Sacks model while they are false under the continuum hypothesis. Usually the proofs are very technical and involve heavy forcing machinery.
The authors have extracted a combinatorial principle that is true in the Sacks model and can be used to give simpler proofs. This principle, which is called CPA, is similar to Martin’s axiom. CPA captures the essence of the Sacks model at least for many cardinal characteristics of the continuum. So it is known that for a “nice” cardinal invariant \(\kappa\), if \(\kappa< {\mathfrak c}\) holds in any forcing extension, then \(\kappa< {\mathfrak c}\) follows already from CPA.
The CPA in its full strength requires some extra definitions that are unnecessary for most of the applications. Therefore the authors introduce the axiom in several approximations. In Chapter 1, they start with the simplest form of the axiom, \(\text{CPA}_{\text{cube}}\), which is based on the notion of a cube in a Polish space. They show a lot of consequences of this axiom. So for instance they show that it implies that every perfectly meager set \(S\subseteq\mathbb{R}\) has cardinality less than \({\mathfrak c}\). The axiom also implies the so-called total failure of Martin’s axiom, which is the following statement: \({\mathfrak c}>\omega_1\) and for every nontrivial forcing \(\mathbb{P}\) satisfying the countable chain condition there exist \(\omega_1\) many dense sets in \(\mathbb{P}\) such that no filter intersects all of them. Further on it is shown that \(\text{CPA}_{\text{cube}}\) is false in the model obtained by adding Sacks reals side by side.
In Chapter 2 the authors investigate the axiom \(\text{CPA}^{\text{game}}_{\text{cube}}\). This axiom is formulated with the help of a covering game for Polish spaces. Among the many applications of this axiom the authors show that \(\text{CPA}^{\text{game}}_{\text{cube}}\) implies that (1) \({\mathfrak c}>\omega_1\) and for every Polish space \(X\) there exists a partition into \(\omega_1\) disjoint closed nowhere dense measure zero sets; (2) \({\mathfrak c}> \omega_1\) and there exists a family \({\mathfrak F}\subseteq [\omega]^\omega\) of cardinality \(\omega_1\) that is simultaneously maximal almost disjoint and reaping.
In Chapter 3 the notion of a cube in a Polish space is generalized to the notion of prism, which gives the axioms \(\text{CPA}_{\text{prism}}\) and \(\text{CPA}^{\text{game}}_{\text{prism}}\). In this and in the two following chapters many applications of these axioms are given. So it is shown that the additivity of Marczewski’s ideal \(s_0\), is equal to \(\omega_1< {\mathfrak c}\).
In Chapter 6, the axiom CPA is formulated and it is shown that this axiom implies all the other versions of the axiom. Finally, in Chapter 7, the authors show that CPA holds in the iterated perfect set model.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03E17 Cardinal characteristics of the continuum
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