Discovering modern set theory. II: Set-theoretic tools for every mathematician.

*(English)*Zbl 0887.03036
Graduate Studies in Mathematics. 18. Providence, RI: American Mathematical Society (AMS). xiii, 224 p. (1997).

The book under review is the second volume of a graduate course in set theory (for the first volume see Zbl 0854.04001). It is aimed at more advanced graduate students and research mathematicians specializing in fields other than set theory. It can be used as a text in the classroom as well as for self-study.

The book contains a short but rigorous introduction to various set-theoretic techniques that have found applications outside of set theory. As the authors mention, their most important criterion for inclusion of an item was frequency of use outside of pure set theory.

The first chapter, “Filters and ideals in partial orders”, has introductory character, but contains also results on compactness, the method of ultrapowers and on nonstandard numbers. The next chapter, “Trees”, considers the question of existence of various sorts of infinite trees – narrow, broad, with no or with many cofinal branches, especially the existence of a Souslin tree/line. The chapter “A little Ramsey theory” deals with different types of colouring/partition problems (with various cardinal, ordinal or topological properties). A finitary variant of a Ramsey theorem is presented, which nevertheless can be proved only by infinitary methods (it is not provable in Peano arithmetic). A short chapter with the important technical “Delta lemma” follows. Extensions of the Fubini theorem, Luzin sets, Sierpinski sets and various other types of thin sets of reals concerning measure and category – including the Borel conjecture – are discussed in the chapter “Applications of the Continuum Hypothesis”. The next two chapters on “Rasiowa-Sikorski theorem” and “Martin’s axiom” present an important technique for proving the existence of objects with a Baire theorem like argument using various partial orders, mainly connected with natural and/or real numbers. This is useful, e.g., if one needs to construct a countable object fulfilling uncountably many conditions. Ultrafilters on natural numbers are studied too. An interesting phenomenon of “Hausdorff Gaps” is still challenging set theoretists. The chapter “Closed unbounded sets and stationary sets” gives an important tool for uncountable constructions. For instance, if along a construction of length \(\omega_1\), on each step \(\alpha\) one fixes/arranges something for a chosen \(\beta < \alpha\) (regressive function) then one can be sure that confinally many times the same \(\beta\) was chosen (Fodor’s theorem). Illustrations from group theory are given. The chapter “Diamond-principle” presents another technique which can be useful when one tries simultaneously to extend countable constructions. A nice illustration of the use of “Diamond” is presented on the construction of a Souslin line/tree. Some of these techniques are absolute in the sense that they are theorems of set theory ZFC, nevertheless some of them, as “Martin’s axiom” or “Diamond” are not, but they are at least consistent with the axioms of ZFC. The chapter “Measurable cardinals” discusses a notion of which we even do not know whether it is consistent with ZFC or not. But it is connected to the important problem whether it is possible to extend Lebesgue measure to a probabilistic measure on all subsets of the reals. Reading the chapter on “Elementary submodels” we can taste the flavour of using model theoretic methods. This is illustrated by theorems from topology. The last chapter studies boolean algebras in general and in particular those which are connected to natural and real numbers. The book closes with an appendix on general topology.

At the end let us quote the authors themselves concerning the style of the book: “…Much of this book is written like a dialogue between the authors and the reader. This is intended to model the practice of creative mathematical thinking, which more often than not takes on the form of an inner dialogue in a mathematician’s mind. You will quickly notice that this text contains many question marks. This reflects our conviction that in the mathematical thought process it is at least as important to have a knack for asking the right question at the right time as it is to know some of the answers. You will benefit from this format only if you do your part and actively participate in the dialogue…”.

The book contains a short but rigorous introduction to various set-theoretic techniques that have found applications outside of set theory. As the authors mention, their most important criterion for inclusion of an item was frequency of use outside of pure set theory.

The first chapter, “Filters and ideals in partial orders”, has introductory character, but contains also results on compactness, the method of ultrapowers and on nonstandard numbers. The next chapter, “Trees”, considers the question of existence of various sorts of infinite trees – narrow, broad, with no or with many cofinal branches, especially the existence of a Souslin tree/line. The chapter “A little Ramsey theory” deals with different types of colouring/partition problems (with various cardinal, ordinal or topological properties). A finitary variant of a Ramsey theorem is presented, which nevertheless can be proved only by infinitary methods (it is not provable in Peano arithmetic). A short chapter with the important technical “Delta lemma” follows. Extensions of the Fubini theorem, Luzin sets, Sierpinski sets and various other types of thin sets of reals concerning measure and category – including the Borel conjecture – are discussed in the chapter “Applications of the Continuum Hypothesis”. The next two chapters on “Rasiowa-Sikorski theorem” and “Martin’s axiom” present an important technique for proving the existence of objects with a Baire theorem like argument using various partial orders, mainly connected with natural and/or real numbers. This is useful, e.g., if one needs to construct a countable object fulfilling uncountably many conditions. Ultrafilters on natural numbers are studied too. An interesting phenomenon of “Hausdorff Gaps” is still challenging set theoretists. The chapter “Closed unbounded sets and stationary sets” gives an important tool for uncountable constructions. For instance, if along a construction of length \(\omega_1\), on each step \(\alpha\) one fixes/arranges something for a chosen \(\beta < \alpha\) (regressive function) then one can be sure that confinally many times the same \(\beta\) was chosen (Fodor’s theorem). Illustrations from group theory are given. The chapter “Diamond-principle” presents another technique which can be useful when one tries simultaneously to extend countable constructions. A nice illustration of the use of “Diamond” is presented on the construction of a Souslin line/tree. Some of these techniques are absolute in the sense that they are theorems of set theory ZFC, nevertheless some of them, as “Martin’s axiom” or “Diamond” are not, but they are at least consistent with the axioms of ZFC. The chapter “Measurable cardinals” discusses a notion of which we even do not know whether it is consistent with ZFC or not. But it is connected to the important problem whether it is possible to extend Lebesgue measure to a probabilistic measure on all subsets of the reals. Reading the chapter on “Elementary submodels” we can taste the flavour of using model theoretic methods. This is illustrated by theorems from topology. The last chapter studies boolean algebras in general and in particular those which are connected to natural and real numbers. The book closes with an appendix on general topology.

At the end let us quote the authors themselves concerning the style of the book: “…Much of this book is written like a dialogue between the authors and the reader. This is intended to model the practice of creative mathematical thinking, which more often than not takes on the form of an inner dialogue in a mathematician’s mind. You will quickly notice that this text contains many question marks. This reflects our conviction that in the mathematical thought process it is at least as important to have a knack for asking the right question at the right time as it is to know some of the answers. You will benefit from this format only if you do your part and actively participate in the dialogue…”.

Reviewer: P.Vojtáš (Košice)