[bibtex key=Hamkins2005:TourlakisBookReview]

Review of George Tourlakis, Lectures in Logic and Set Theory, volumes 1 and 2, Cambridge studies in advanced  mathematics, vol. 83.  Cambridge University Press, Cambridge, UK, 2003. This is a detailed two-volume development of mathematical logic and set theory, written  from a formalist point of view, aimed at a spectrum of  students from the third-year undergraduate to junior  graduate level. Volume 1 presents the heart of mathematical  logic, including the Completeness and Incompleteness theorems along with a bit of computability theory and accompanying ideas. Tourlakis aspires to include “the absolutely essential topics in proof, model and recursion theory” (vol. 1, p. ix). In addition, for the final third of the volume, Tourlakis provides a proof  of the Second Incompleteness Theorem “right from Peano’s axioms,…gory details and all,” which he conjectures “is the only complete proof in print [from just Peano arithmetic] other than the one that was given in Hilbert and Bernays (1968)” (vol. 1, p. x). In the opening
page of Chapter II, Tourlakis provides a lucid explanation of the proof in plain language, before diving into the details and emerging a hundred pages later with the provability predicate, the derivability conditions and a complete proof. Tempering his formalist tendencies, Tourlakis speaks “the formal language with a heavy `accent’ and using many `idioms’ borrowed from `real’ (meta)mathematics and English,” in a mathematical argot (vol. 1, p. 39). In his theorems and proofs, therefore, he stays close to the formal language without remaining inside it.

But let me focus on volume 2, a stand-alone development of axiomatic set theory, containing within it a condensed version of volume 1. The book emphasizes the formal
foundations of set theory and, like the first volume, gives considerable attention to the details of the elementary theory. Tourlakis is admirably meticulous in maintaining
the theory/metatheory distinction, with a careful explanation of the role of inductive arguments and constructions in the metatheory (vol. 2, p. 20) and a correspondingly precise treatment of axioms, theorems and their respective schemes throughout. What is more, he sprinkles the text with philosophical explanations of the theory/metatheory interaction, giving a clear account, for example, of how it is that we may use apparently set theoretic arguments in the metatheory without circularity (vol. 1, p. 10-12). After developing the logical background, he paints the motivating picture of the cumulative hierarchy, the process by which we imagine sets to be built, with Russell’s paradox as a cautionary tale. In Chapter III, the axioms of set theory march forward in succession. He presents them gradually, motivating them from the cumulative hierarchy and deriving consequences as they appear. This treatment includes the Axiom of Choice, which he motivates, impressively, by developing Goedel’s constructible universe $L$ sufficiently to see that the Axiom of Choice holds there. Later, he revisits the constructible universe more formally, and by the end of the book his formal set theoretic development encompasses even the sophisticated topic of forcing. The book culminates in Cohen’s relative consistency proof, via forcing, of the failure of the Continuum Hypothesis.

Interestingly, Tourlakis’ version of ZFC set theory, like Zermelo’s,  allows for (without insisting on) the existence of urelements, atomic objects that are not sets, but which
can be elements of sets. His reason for this is philosophical and pedagogical: he finds “it extremely counterintuitive, especially when addressing undergraduate audiences, to tell them that all their familiar mathematical objects — the `stuff of mathematics’ in
Barwise’s words — are just perverse `box-in-a-box-in-a-box\dots’ formulations built from an infinite supply of empty boxes” (vol. 2, p. xiii). The enrichment of the theory to allow
urelements requires only minor modifications of the usual  ZFC axioms, such as the restriction of Extensionality to the sets and not the urelements. The application of the
definition $a\subseteq b\iff\forall z(z\in a\implies z\in b)$ even when $a$ or $b$ are urelements, however, causes some peculiarities, such as the consequence that urelements are subsets of every object, including each other. Consequently, the axiom asserting that the urelements form a set (Axiom III.3.1), is actually deducible via the
Comprehension Axiom from Tourlakis’ version of the Power Set Axiom, which asserts that for every object $a$ there is a set $b$ such that $\forall x(x\subseteq a\implies x\in
b)$, since any such $b$ must contain all urelements.

At times, the author employs what some might take as an exaggerated formal style. For example, after introducing the Pairing Axiom, stating that for any $a$ and $b$ there
is $c$ with $a\in c$ and $b\in c$, he considers Proposition  III.5.3, the trivial consequence that $\{a,b\}$ is a set. His first proof of this is set out in eleven numbered
steps, with duly noted uses of the Leibniz axiom and modus ponens. To be sure, he later adopts what he calls a “relaxed” proof style, but even so, in the “Informal”
Example III.9.2, he fills a page with tight reasoning and explicit appeals to the deduction theorem, the principle of auxiliary constants and more, to show merely that if $x$ is
a set and $x\subseteq\{\emptyset\}$, then $x=\emptyset$ or $x=\{\emptyset\}$. Similar examples of formality can be found on pages 118, 120, 183-184 and elsewhere in volume 2, as well as volume 1.

The preface of volume 2 explains that the book weaves a middle path between those set theory books that merely build set-theoretic tools for use elsewhere and those that
aim at research in set theory. But I question this assessment. Many of the topics constituting what I take to be the beginnings of the subject appear only very late in
the book. For example, the von Neumann ordinals appear first on page 331; Cantor’s theorem on the uncountability of $P(\omega)$ occurs on page 455; the Cantor-Bernstein theorem appears on page 463; the definitions of cardinal successor and $\aleph_\alpha$ wait until page 465; and the definition of cofinality does not appear until page 478, with regular and singular cardinals on page 479. Perhaps it was the elaborate formal development of the early theory that has pushed this basic part of set theory to the end of the book. This may not be a problem, but I worry that students may wrongly understand these topics to constitute “advanced” set theory, when surely the opposite is true. Furthermore, many other elementary topics, which one might expect to find in a set theory text aimed in part at graduate students, do not appear in the text at all. This includes closed unbounded sets, stationary sets, $\omega_1$-trees (such as Souslin trees or Kurepa trees), Borel sets, regressive functions, Martin’s axiom, the
diamond principle and even ultrafilters. Large cardinals are not mentioned beyond the inaccessible cardinals. The omission of ultrafilters is particularly puzzling, given
the author’s claim to have included “all the fundamental tools of set theory as needed elsewhere in the mathematical sciences” (vol. 2, p.~xii). Certainly ultrapowers are one
of the most powerful and successful such tools, whose fundamental properties remain deeply connected with logic.

In the final chapter, the author provides a formal account of the foundations of forcing, with useful explanations again of the important theory/metatheory interaction
arising in connection with it. Because his account of forcing is based on countable transitive models, some set theorists may find it old-fashioned. This way of forcing
tends to push much of the technique into the metatheory, which Tourlakis adopts explicitly (vol. 2, p. 519), and can sometimes limit forcing to its role in independence
results. A more contemporary view of forcing makes sense within ZFC of forcing over $V$, for example via the Boolean-valued models $V^{\mathbb B}$, and allows one
sensibly to discuss the possibilities achievable by forcing over any given model of set theory.

Despite my reservations, I welcome Tourlakis’ addition to the body of logic texts. Readers with a formalist bent especially will gain from it.