This book makes me feel old. I was once very keen on set theory, spending several years in undergraduate school and even some courses in graduate school studying it, as usual in tandem with mathematical logic. That was back in the late 1970s and early 1980s. My, oh my, how times have changed.
The book under review, while truly an introduction to the beautiful subject of cardinal arithmetic, as the title claims, starts off with Z(ermelo) F(raenkel) C(hoice — as in the Axiom of Choice) set theory, and, to be sure, in the first hundred or so pages we come across material still quite familiar to an old(ish) fogey like me. In a very succinct and compact (and effective) fashion the famous axiom list for ZFC is presented on pp. 6–7, and then it’s off to the races: great fun. We reach the Erdös-Rado petition theorem on p.96. Say the authors: “Part one [of the book’s three parts], which is contained in Chapter 1, describes the classical cardinal arithmetic due to Bernstein, Cantor, Hausdorff, König, and Tarski. The results were found in the years between 1870 and 1930.”
Whoa! So my training is set theory was already out of date in the 1970s! Well, fair enough, I guess. What about Tarski, though, whose name I kept hearing when I was in school? Well, come to think of it, I guess that his later career was concerned more with mathematical logic than with set theory proper, so my hearing about his work as being of relatively recent vintage is not inconsistent with the proposition that qua set theory I might actually already have been something of a dinosaur. (And I guess the set theorists at my school (UCLA) were indeed already pretty keen on avant garde stuff by Solovay and others, quite out of my league as a fellow traveler and dilettante.)
But now, thirty or so years later (tempus fugit), it turns out that the 1970s were in fact a watershed of sorts in set theory: again the authors: “Part two, which is Chapter 2, characterizes the development of cardinal arithmetic in the seventies, which was led by Galvin, Hajnal, and Silver.” And so it is that Chapter 2 is about the Galvin-Hajnal Theorem, which is a very strong result belonging to the context of the singular cardinal problem. Here’s something from the introduction to Chapter 2: “The … singular cardinal problem consists of the description of the possible … value of the gimel function at the argument [aleph-eta], for singular cardinals [aleph-eta]. An estimate for this cardinal power is given [under certain circumstances] by the Galvin-Hajnal theorem … The centre of our investigations will be the Galvin-Hajnal formula, from which all other on cardinals in this chapter will follow …”
Well, there you have it. We’re only on p.103, and the waters begin to rise. We just had Erdös-Rado to play with and even as late as p.102 we encounter something more or less prosaic (or, in set theorists’ parlance, at least weakly accessible — Hah!): “Prove that every weakly compact cardinal is inaccessible [!]” But on p.103 we begin to speed through the 1970s. It is clear as a bell that this book is not meant for fellow travelers and dilettantes. It’s very, very serious set theory, even if it’s defined to be an introduction to the subject.
This is underscored in spades by the third part of the book, Chapters 3–9, which, the authors describe as presenting “the fundamental investigations in pcf [= ‘possible cofinalities’]-theory which has been developed by S. Shelah to answer the questions left open in the seventies.” Indeed, they go on to say that “[a]ll theorems presented in Chapter 3 and Chapters 5 to 9 are due to Shelah, unless otherwise stated.” So, yes indeed, we now come to the work of Saharon Shelah, who is still going very strong (I ran across some of his stuff on the arXiv only a week or so ago), already having on the order of 900 (yes, 900!) papers to his name. We are smack-dab in the middle of modern set theory.
So, this Introduction to Cardinal Arithmetic takes the reader from the birth of the subject in 1872 (with the early work of Georg Cantor) to, well, pretty much today, or, if not, then late yesterday afternoon (the book’s original publication date being 1999: this is a Modern Birkhäuser Clasics re-issue). As I noted earlier, it’s very serious business, and the reader should really want to become a set theorist himself, if he’s to go any real distance with this book. But there are lots of exercises (that look pretty sporty to me), and the authors have taken great pains to prove everything very carefully and thoroughly. It’s obviously a fine source for those inclined to go this route.
Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.