This is a text on the upper division undergraduate or lower graduate level meant to introduce the student to set theory, especially to consistency proofs of CH and of \(\neg\)CH. The approach, rooted in the authors’ mathematical research interests, is somewhat idiosyncratic, and the bulk of the review will sketch the somewhat non-standard approaches of the book. The two authors have different styles and different concerns (Smullyan all by himself has two different rhetorical styles, the informal style of his popular books, and a more standard technical style, both of which are evident here), so added to the idiosyncracies are disjunctive rhetorics.
Smullyan’s work has focused on variations of induction and fixed points; Fitting has done much of his work on modal logic. Their goal in this book seem to be to find the best framework for set theory with the particular goal of understanding why CH is independent, and the decisions they have made reflect their research interests.
They have chosen a class set theory (NBG) for their base, abandoning the usual easy reliance on first order logic (especially model theory) that make using ZF so convenient (although this is not too great an obstacle). Smullyan’s interests in superinduction and fixed points leads to an extensive development of terminology so that various inductive arguments can be unified. And, most radically, Fitting’s interest in modal logic leads to a development of forcing by means of modal logic. (This is not surprising. Way back in 1969, in his book: Intuitionistic logic, model theory and forcing (1969; Zbl 0188.32003), M. Fitting used intuitionism to explicate forcing.)
For the curious reader, here is a rough description of how modal logic is used. An operator \([[ \cdot]]\) on sentences is introduced. Very roughly speaking, “\([[A]]\)” means “necessarily possibly \(A\)”. A forcing condition \(p\) is a possible world in a frame (we are in Kripke semantics here), and rather than talking about \(p\) forcing \(A\) we talk about \(p \Vdash [[A]]\). For the cognosceti, the semantics used is S4, i.e., pre-orders.
It works, and rather prettily too, but of course is very far from the standard treatment of forcing. Will this approach prove so superior in elegance and utility to the more standard approach through partial orders (using either first-order countable transitive models or Boolean-valued models) that it will supplant the standard approach? In the long run one never knows. In the short run, the answer clearly is no.
Which raises a second question: does the approach here transfer easily to the current standard approach so that a student familiar with one can understand the other? I’m not sure. Since the semantics is based on pre-orders the translation is not that difficult to make (in fact some technicalities about, say, adding a random real, are eased). A chapter (“Constructing classical models”) makes the connection explicit via generic sets. So maybe the transfer can be made easily.
And finally, a third question: can the notion of modal logic be meaningful (whatever that means) to a student when developed so very quickly towards the end of a book? I honestly don’t know.
As for the rest of the book, there is a standard development of ordinals and cardinals (with the addition of class choice functions, since classes are available), and a careful development of the notion of model of set theory and of \(L\). Their models are standard – “\(\epsilon\)” denotes \(\epsilon\) – so some of the arcana of semantics, e.g., definitions of satisfaction and truth, can be avoided. Even so, they provide such general notions for a language with one unary predicate.
Missing are the topics in combinatorics and the applications of set theory to other fields of mathematics that have been a prominent part of most recent set theory books at this level.