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Consequences of Martin’s axiom. (English) Zbl 0551.03033

Cambridge Tracts in Mathematics, 84. Cambridge etc.: Cambridge University Press. XII, 325 p. £27.50; $ 54.50 (1984).
Martin’s axiom (MA) has been around since late 1960’s. The axiom itself was extracted from various consistency proofs following the consistency of the Souslin Hypothesis due to Solovay and Tennenbaum. Martin’s axiom is consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZFC) and it has the status of an additional set-theoretic hypothesis. Although it follows from the continuum hypothesis (CH), it is consistent with the negation of the continuum hypothesis (\(\neg CH)\) and most of its interesting consequences can be derived in \(ZFC+\neg CH.\) The number of theorems that follow from it makes MA one of the most powerful methods available to the non-logician, of showing the consistency of undecidable propositions. It has been applied in topology, algebra, measure theory, combinatorics and other relevant fields of mathematics and it implies e.g. the existence of a non-metrizable normal Moore space or of a non-free Whitehead group, just to mention some of many of its consequences.
The book under review collects into one volume most of the propositions that can be derived from MA and have been known so far. The discussion is centered around the three cardinal numbers m, \(m_ K\) and p related to two versions of Martin’s axiom and a related combinatorial principle P(\(\kappa)\). The cardinal m is the least among the cardinals \(\kappa\) such that Martin’s axiom MA(\(\kappa)\) is false and \(m_ K\) is the least cardinal such that the version MAK(\(\kappa)\) for partial orderings satisfying the Knaster condition fails. It can be shown that \(\omega_ 1\leq m\leq m_ K\leq p\leq 2^{\omega}.\) Several equivalent characteristics of the cardinals m, \(m_ K\) and p are given in Chapter 1 and the three subsequent chapters discuss each one of them. They present a systematic discussion of the propositions that follow from cardinal assumptions \(p>\omega_ 1\), \(m_ K>\omega_ 1\), and \(m>\omega_ 1\) to combinatorics, descriptive set theory, measure theory and topology. Among many other problems, Chapter 2 deals with the normal Moore space conjecture, Chapter 3 shows that \(MA+CH\) implies that there is a non-free Whitehead group and Chapter 4 deals with the relation between MA and separability of topological spaces. These chapters include a large number of other results related to the respective cardinals. It would be difficult to make a representative selection of them and not to forget any important one.
The book is well-written and accessible to anyone working in a field where Martin’s axiom is applicable. The text is self-contained, accompanied by two appendices supplying all relevant definitions and facts from set theory, general topology, measure theory and functional analysis. The book is recommendable to anyone interested in mathematical results related to Martin’s axiom.
Reviewer: P.Štěpánek

MSC:

03E50 Continuum hypothesis and Martin’s axiom
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E15 Descriptive set theory