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Cardinal arithmetic. (English)
Oxford Logic Guides. 29. Oxford: Clarendon Press. xxxi, 481 p. \sterling 65.00 (1994).
The work presented in this book could be thought of as extensions of {\it J. Silver’s} result on the singular cardinal problem [in Proc. Int. Congr. Math., Vancouver 1974, Vol. 1, 265-268 (1975; Zbl 0341.02057)]. It culminates in the result presented in chapter IX, of which the simplest case is: (1) If $2^{\aleph_0} < \aleph_ω$, then $\aleph_ω^{\aleph_0} < \aleph_{ω_4}$. In one sense, statement (1) distorts what is being presented: a major thesis of the book (the naturalness thesis) is that the cardinal powers studied since Cantor are crude measures, which need to be refined if we are to progress. The refined notions, introduced in the first two chapters, are the set of possible cofinalities, $\text {pcf} (a)$, of reduced powers of a set $a$ of cardinals; and for a singular cardinal $λ$, $\text {pp} (λ)$, which is the supremum of $\text {pcf} (\{λ_i \mid i < \text {cf} λ\})$ for any sequence of regular cardinals $\langle λ_i \mid i < \text {cf} λ\rangle$ with supremum $λ$. So the real theorem presented in chapter IX is (2): $\text {pp} (\aleph_ω) < \aleph_{ω_4}$, which implies (1). It is shown that the usual cardinal powers can be derived from the values of $\text {pp} (λ)$ for all singular $λ$, but not vice versa; and part of the thesis that (2) is the natural result lies in the fact that it is a theorem of ZFC, in contrast with (1) which requires the extra hypothesis $2^{\aleph_0} < \aleph_ω$. It is argued that this means that the results might have meaning to Cantor, if he were to return to read them, in contrast with results such as forcing results or results on inner models. Further support derives from applications of the results, which are given in many chapters: there are applications to colouring theorems, results on the existence of Jonsson algebras, of narrow Boolean algebras and entangled linear orders, of $λ$-free not free Abelian groups, of almost disjoint families, and of $L_{\infty, λ}$-equivalent non-isomorphic models of singular cardinality $λ$. These applications are mostly extensions of known results to further singular cardinals. The book consists of nine main chapters, which could be thought of as a collection of nine research papers, with to further papers as appendices. There is a substantial introduction, including an author’s review, detailed annotated contents of the chapters and appendices, a summary of the history of the subject (which is almost exclusively due to the author since about 1980), and a section “Could Cantor really have read it or what the reader is assumed to know”. There is no index, but there is a 25 page “analytical guide” which gives the location of many of the definitions; and there is an “annotated content of continuations”, which describes the content of ten further papers of the author which continue the work presented (which all date prior to December 1989).
F.R.Drake (Leeds)