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**Introduction to cardinal arithmetic.**
*(English)*
Zbl 0930.03053

Birkhäuser Advanced Texts. Basel: Birkhäuser. vii, 304 p. (1999).

The authors of this monograph-text intend to bring the reader from the basics of cardinal number theory within ZFC to the current state of knowledge of the (infinite) cardinal function \(\kappa^{\text{ cf}(\kappa)}\), again within ZFC. For the latter, the pcf-theory of Shelah is pivotal and chapters 3-9, some 150 pages, are devoted to the work of Shelah. We give a brief description of the text’s contents.

Chapter 1 with its extensive exercises gives a thorough introduction to cardinals and cardinal arithmetic. It starts with the axioms of ZFC and ends with the Erdős-Rado Partition Theorem. Chapter 2 introduces and proves the Galvin-Hajnal formula, which provides a non-trivial upper bound for \(\aleph_{\eta}^{\text{ cf}(\aleph_{\eta})}\) when \(\aleph_{\eta}\) is singular and of uncountable cofinality. A famous consequence of Galvin-Hajnal is Silver’s theorem that for \(\kappa\) singular and of uncountable cofinality, if \(2^{\mu}=\mu^+\) for all infinite \(\mu<\kappa\), then \(2^{\kappa}=\kappa^+\). Chapters 3-9 are about Shelah’s pcf-theory. Chapter 3, called Ordinal Functions, introduces the theory’s terminology and basic results. Chapter 4, Approximation Sequences, is about certain sequences of elementary substructures of \(H(\Theta)\), the sets that are hereditarily of cardinality smaller than \(\Theta\). These sequences are an important technical tool. Chapters 5-7 cover the more profound aspects of pcf-theory. In Chapter 8, the rewards of this hard work are obtained. Amongst other applications, we learn that for \(\delta\) a limit ordinal \(\aleph_\delta^{|\delta|}<\max\{\aleph_{|\delta|^{+4}}, (2^{|\delta|})^+\}\). In particular, \(2^{\aleph_0}<\aleph_{\omega}\) implies \(\aleph_{\omega}^{\aleph_0}< \aleph_{\omega_4}\). Chapter 9, the final chapter, is on the cardinal function pp(\(\lambda\)).

The first four chapters contain many exercises that broaden and apply material from the body of the text. A small number are of the “prove lemma 1.45” variety, but it appears that progress through the meat of the text does not rely upon the exercises. There is an extensive list of symbols and a decent index.

Chapter 1 with its extensive exercises gives a thorough introduction to cardinals and cardinal arithmetic. It starts with the axioms of ZFC and ends with the Erdős-Rado Partition Theorem. Chapter 2 introduces and proves the Galvin-Hajnal formula, which provides a non-trivial upper bound for \(\aleph_{\eta}^{\text{ cf}(\aleph_{\eta})}\) when \(\aleph_{\eta}\) is singular and of uncountable cofinality. A famous consequence of Galvin-Hajnal is Silver’s theorem that for \(\kappa\) singular and of uncountable cofinality, if \(2^{\mu}=\mu^+\) for all infinite \(\mu<\kappa\), then \(2^{\kappa}=\kappa^+\). Chapters 3-9 are about Shelah’s pcf-theory. Chapter 3, called Ordinal Functions, introduces the theory’s terminology and basic results. Chapter 4, Approximation Sequences, is about certain sequences of elementary substructures of \(H(\Theta)\), the sets that are hereditarily of cardinality smaller than \(\Theta\). These sequences are an important technical tool. Chapters 5-7 cover the more profound aspects of pcf-theory. In Chapter 8, the rewards of this hard work are obtained. Amongst other applications, we learn that for \(\delta\) a limit ordinal \(\aleph_\delta^{|\delta|}<\max\{\aleph_{|\delta|^{+4}}, (2^{|\delta|})^+\}\). In particular, \(2^{\aleph_0}<\aleph_{\omega}\) implies \(\aleph_{\omega}^{\aleph_0}< \aleph_{\omega_4}\). Chapter 9, the final chapter, is on the cardinal function pp(\(\lambda\)).

The first four chapters contain many exercises that broaden and apply material from the body of the text. A small number are of the “prove lemma 1.45” variety, but it appears that progress through the meat of the text does not rely upon the exercises. There is an extensive list of symbols and a decent index.

Reviewer: J.M.Plotkin (East Lansing)

### MSC:

03E10 | Ordinal and cardinal numbers |

03E04 | Ordered sets and their cofinalities; pcf theory |

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