[bibtex key=Hamkins2000:BookReviewKanamori]

**Akihiro Kanamori**. *The Higher Infinite*. Large cardinals, stealing upwards through the clouds of imagined limitation like the steel skyscrapers of a ever-growing set-theoretic skyline, reach towards the stratosphere of Cantor’s *absolute*. In this century we have axiomatized larger and larger notions of infinity, and as we live amongst these giants, the formerly tall now seem small. Weakly inaccessible cardinals, for example, first considered by Hausdorff as a natural transfinite limit of set-theoretic operations, now occupy a floor at the entryway to the large cardinal hierarchy. In time over the past century we had Mahlo cardinals, strongly inaccessible cardinals, measurable cardinals, indescribable cardinals, weakly-compact cardinals, strongly-compact cardinals, super-compact cardinals, huge cardinals, almost huge cardinals, superhuge cardinals, and so on. And while when it comes to naming these enormous magnitudes, words have perhaps failed us, the mathematics is perfectly precise and fascinating.

Professor Kanamori has written—beautifully so—the book we large cardinal set-theorists have been lacking, a book spanning the possibilities from inaccessible to superhuge cardinals and beyond, a book full of historical insight, clear writing, interesting theorems and elegant proofs. This book is destined to become, if it has not already become, the standard reference in its field.

Finding that “a genetic account through historical progression…provides the most coherent exposition of the mathematics and holds the key to any epistemological concerns,” (p. XI) Kanamori weaves a historical perspective into the mathematics, deepening our understanding and appreciation of it. He sprinkles the text with quotations of Gödel and others, giving their mathematical-philosophical views on the mathematical developments. The introduction stands alone as a non-technical essay introducing the entire subject. From there, Kanamori begins with the smaller large cardinals, inaccessible and Mahlo cardinals, and then moves in time up to the strongest hypotheses.

So let me begin to explain a little about large cardinals. A cardinal $\kappa$ is *inaccessible* when it cannot be constructed from smaller cardinals, so that first, it is not the supremum of fewer than $\kappa$ many cardinals each of size less than $\kappa$ (as, for example, $\aleph_\omega=\sup_n\aleph_n$ is), and second, it cannot be reached by the power set operation in the sense that whenever $\delta$ is smaller than $\kappa$ then $2^\delta$ is also smaller than $\kappa$. It is relatively straightforward to show that if $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC. In particular, if $\kappa$ is the least inaccessible cardinal, then $V_\kappa$ will be a model of ZFC in which there are no inaccessible cardinals. So it is relatively consistent with ZFC that there are no large cardinals at all. Furthermore, since the mere existence of an inaccessible cardinal provides a full model of ZFC, we cannot hope even for a *relative* consistency result of the form “If ZFC is consistent, then so is ZFC $+$ there is an inaccessible cardinal” (in the manner of results proved for the Continuum Hypothesis and the Axiom of Choice), for then the theory “ZFC $+$ there is an inaccessible cardinal” would imply its own consistency, contrary to Gödel’s Incompleteness Theorem. In short, the consistency strength of the existence of an inaccessible cardinal is greater than that of ZFC alone. At first glance, then, the logical status of the existence of even the smallest of the large cardinals is a bit startling: we can’t prove they exist; it is consistent that they don’t exist; and we can *prove* that we cannot prove that their existence is relatively consistent. What, then, is the point of them?

The point is that such a transcendence over ZFC in consistency strength is exactly what we want and what we need. In the decades since the invention of Cohen’s forcing technique, set theorists have set marching an infinite parade of independence results; indeed, it often seems as though almost all the interesting set-theoretic questions are independent of our ZFC axioms. We all know now that the cardinality of the set $\mathbb{R}$ of reals can be $\aleph_1$ or $\aleph_2$ or $\aleph_{1776}$ or $\aleph_{\omega+1776}$ or any cardinal you like within reason, and this unfinished nature of ZFC when it comes to basic set theoretic questions is the norm. We have learned in this sense that ZFC is a weak theory. The large cardinal axioms provide strengthenings of it, strengthenings which are fundamentally different from the strengthenings of ZFC provided by the Continuum Hypothesis, the Generalized Continuum Hypothesis, Souslin’s Hypothesis, Martin’s Axiom and many of the other principles that we know to properly extend ZFC, in that large cardinals transcend even the consistency strength of ZFC. The large cardinal hierarchy, therefore, in addition to its intrinsic mathematical interest, provides a natural structure which can be used to gauge the consistency strength of general mathematical propositions.

Let me give one example. Almost all mathematicians are familiar with Vitali’s construction of a non-Lebesgue measurable set of reals and furthermore believe that the construction makes an essential use of the Axiom of Choice AC. But what does this mean exactly? The impossibility of removing AC from the Vitali construction is equivalent to the consistency (without AC) that every set of reals is Lebesgue measurable. Now of course we need some choice principle to develop a satisfactory theory of Lebesgue measure at all, so let us keep in the base theory the principle of Dependent Choices DC, which allows us to make countably many choices in succession. Thus, we are led to consider the consistency of the theory $T=$ “ZF + DC + every set of reals is Lebesgue measurable”. Solovay [65] proved that if ZFC is consistent with the existence of an inaccessible cardinal, then $T$ is consistent; that is, if inaccessible cardinals are consistent, then we are perfectly correct in believing that you cannot remove AC from Vitali’s construction. Since most mathematicians already believed this conclusion, Solovay’s use of an inaccessible cardinal was widely seen as a defect in his argument. But Shelah [84] exploded this criticism by proving conversely that if $T$ is consistent, then so is the existence of an inaccessible cardinal. That is, the two theories are *equiconsistent*, and we should be exactly as confident in the consistency of inaccessible cardinals as we are in our belief that Vitali’s use of AC is essential.

After the beginnings, Kanamori moves swiftly through a chapter on partition properties, weak compactness, indiscernibles and $0^\sharp$, before moving into a longer chapter on forcing and sets of reals, in which he introduces forcing, Lebesgue measurability and topics from descriptive set theory. Next, in Chapter Four, he approaches measurability from the direction of saturated ideals, including such topics as Prikry forcing, iterated ultrapower embeddings, the inner model $L[\mu]$, $0^\dagger$ and, curiously, a chess problem for the solution of which he will pay a small prize. The strongest hypotheses appear in Chapter five along with the combinatorial backup needed to support them. Kanamori concludes in Chapter six with the Axiom of Determinacy, giving such connections to large cardinals as can be easily given, and, whetting the appetite of the eager student, surveying the more recent, more difficult, and more amazing results.

Kanamori’s book has already been translated into Japanese by S. Fuchino, and judging by the graduate students I saw last year in Japan pouring over it, the translation seems destined to create a new generation of large cardinal set theorists in Japan.

I do have one reservation about Kanamori’s book, namely, that he didn’t include much material on the interaction between forcing and large cardinals. Admittedly, this being the focus of much of my own work, I harbor some bias in its favor, but the topics of forcing and large cardinals are two major set theoretic research areas, and the intersection is rich. It would have been relatively easy for Kanamori to include a presentation, for example, of the landmark Laver preparation, by which any supercompact cardinal $\kappa$ becomes indestructible by $\kappa$-directed closed forcing. And Laver’s result is really just the beginning of the investigation of how large cardinals are affected by forcing. I trust that much of this work will appear in volume II.

My overall evaluation is entirely positive, and I recommend the book in the strongest possible terms to anyone with an interest in large cardinals. I can hardly wait for the subsequent volume!

[84] Saharon Shelah, “Can you take Solovay’s inaccessible away?” IJM 48 (1984), 1-47.

[65] Robert M. Solovay, “The measure problem,” NAMS 12 (1965), 217.

I always find the analogy of large cardinals to omega and N and that Vomega is a model of ZF- axiom of infinity helpful. Nice review.

Very nice review!

(You should note the proper spelling is Lebesgue, with a G, though).

Thanks very much. I have corrected.