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Handbook of the History of Logic, Volume 6: Sets and Extensions in the Twentieth Century

Dov M. Gabbay, Akihiro Kanamori, and John Woods, editors
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Damir Dzhafarov
, on

The volume under review is a nice addition to the “Handbook of the History of Logic” series, though only about half of it is in any real sense a historical account of set theory, its other half being more of a survey of some of the subject’s present-day directions. But that is what makes this work unique and interesting. Set theory was once a subject that appealed to mathematicians universally, but like other disciplines it has developed increasingly towards the technical and esoteric. Here that development is put in a historical and mathematical context, helping to elucidate not only the subject’s accomplishments, but also its trends and goals. As a result, students of mathematics, amateur and professional alike, should find the essays in this volume instructive and insightful.

The first two chapters (essays) are perhaps of the broadest general interest, being the most expository. Indeed, parts of both chapters should be accessible to senior undergraduate students, especially those motivated enough to look some things up on Wikipedia. These chapters are wonderful historical surveys, the first covering the development of the subject from its inception in the study of the continuum up through the invention of forcing, the second focusing on the continuum problem in detail.

The first chapter, by A. Kanamori, introduces us not only to many of the key players, from Cantor to Cohen (as the title says), but also to many fundamental set-theoretic concepts, from well-orderings and ordinals through Zermelo’s axioms and the cumulative hierarchy to large cardinals and consistency. Thus, although the text gets technical enough in places — especially in the latter sections leading up to forcing — to make reading it in its entirety by no means effortless, it does provide a good informal introduction for anyone looking for an overview of the subject or intending to study it further. The chapter on the continuum, by J. Steprāns, documents the emergence of the continuum hypothesis as a central question in the early part of the 20th century, and discusses many of the efforts to settle it and the new ideas these efforts gave rise to. It concludes, appropriately enough, with a look to the future, at some of the questions that have taken, or may take, the place of the continuum hypothesis in the study of cardinal invariants of the continuum. It is fitting (and a testament to the inexorable march of mathematical progress) that one of the questions highlighted by the author here, about the consistency of \(\mathfrak{p} < \mathfrak{t}\), has been resolved since this volume’s publication, by Malliaris and Shelah.

The next chapter deals with infinite combinatorics, followed by two chapters on large cardinals and their uses in forcing and inner models, respectively. These are definitely more advanced, but it would be unfair to contrast them with the previous general chapters and call them specialized since the topics they deal with are so integral to modern set theory. There is still plenty of friendly, informal discussion. The chapter on combinatorics, by J. A. Larson, is a thorough look at the fascinating world of partition, ordering, and matching problems in infinite cardinalities, problems that are often simple to state yet surprisingly difficult to study. These have spawned a great number of new techniques in set theory and pure combinatorics, and many of these are described here. Alas, not mentioned are some of the investigations of combinatorial principles in recursion theory and reverse mathematics, which are in many ways complementary to the set-theoretic ones described here. But given the author’s focus and the amount of material covered this is no major omission. The other two chapters, by A. Kanamori and W. J. Mitchell, respectively, are equally interesting. In parts they survey topics that are rarely accessible to individuals without degrees in logic, maybe even set theory specifically, including some relatively recent results.

The chapter on determinacy by P. B. Larson underscores the importance in set theory, and in logic generally, of conceiving of certain arguments as games and analyzing winning strategies for players. We are led through the history of the various determinacy results, from open games to Borel games, and then shown relationships between stronger forms of determinacy and other set-theoretic constructs, such as the existence of Woodin cardinals. The subsequent chapter by M. Kojman surveys singular cardinals, and some of the powerful ways in which they have been used in combinatorics, topology, model theory, and elsewhere.

This is followed by a chapter on alternative axiomatizations of set theory by M. R. Holmes, T. Forster, and T. Libert, i.e., on alternatives to the familiar Zermelo-Fraenkel axioms. The authors give a brief account of type theory, and then present several alternative set theories, including new foundations; several systems of so-called positive set theory, which aim to at least partially restore the comprehension axiom to its original formulation as (∃x)(∀y)[y ∈ x ↔ φ(y)] by restricting φ to only positive formulas; and finally, systems motivated by nonstandard analysis. One does not get a very good sense of why one system should be preferable to another, but perhaps this is because, as the author points out, no system has any real advantage in terms of the mathematics it can develop. But the chapter is a good reminder of set theory’s foundational roots, and that these remain firm.

The volume concludes with chapters on three topics that are perhaps less central within contemporary set theory, but no less fascinating. The chapter by J. Bell takes us further into type theory, leading us from its origins in set theory and the relationship of types to sets to its modern position closer to category theory and the relationship of types to categories. This is picked up in the next chapter by J.-P. Marquis and G. E. Reyes with an account of categorical logic which, as a foundational system quite separate from the alternative set theories discussed above, provides a purely algebraic framework for interpreting (and extending) first-order logic. Lastly, F. Kamareddine, T. Laan, and R. Constable take us through Russell’s ramified type theory, focusing on how orders, which Whitehead and Russell developed alongside types in the Principia, can serve as the basis of a foundation for computer science. Thus we see some of set theory’s oldest ideas meeting and combining with some of its newest, and are left with an impression of the subject (if, indeed, everything here can be regarded as part of a single subject) as dynamic and robust.

This is an impressive work, with obvious appeal for logicians, philosophers, and computer scientists. All the contributions are extremely well-written and largely self-contained, certainly enough to allow one to get a reasonable sense of the main ideas presented. It can easily be read for pleasure and reference alike, and one can imagine that over time it will become a favorite in certain circles, if only for its extraordinary achievement of collecting in one place so many wonderful commentaries on such a wide spectrum of topics. If ever there were doubts about the vitality of set theory going into the 21st century, this volume puts them squarely to rest.

Damir Dzhafarov is a Post-doctoral Fellow at the University of California in Berkeley.

Set Theory from Cantor to Cohen, by Akihiro Kanamori
History of the Continuum in the 20th Century, by Juris Steprans
Infinite Combinatorics, by Jean A. Larson
Large Cardinals with Forcing, by Akihiro Kanamori
Inner Models for Large Cardinals, by William J. Mitchell
A Brief History of Determinacy, by Paul B. Larson
Singular Cardinals: From Hausdorff's Gaps to Shelah's pcf Theory, by Menachem Kojman
Alternative Set Theories, by M. Randall Holmes, Thomas Forster, and Thierry Libert
Types, Sets, and Categories, by John L. Bell
The History of Categorical Logic: 1963-1977, by Jean-Pierre Marquis and Gonzalo E. Reyes
Russell's Orders in Kripke's Theory of Truth and Computational Type Theory, by Fairouz Kamareddine, Twan Laan, and Robert Constable