For the London Review of Books, A.W. Moore, editor of a collection of essays on infinity, compares David Foster Wallace’s Everything and More: A Compact History of Infinity with two other books on the subject and concludes that Wallace is wrong:
As for Wallace’s book, the less said, the better. It’s a sloppy production, including neither an index nor a table of contents, and after a while his breezy style grates. No one who is unfamiliar with the ideas behind his dense, user-unfriendly mathematical expositions could work their way through them to gain any insight into what he is talking about. Worse, anyone who is already familiar with these ideas will see that his expositions are often riddled with mistakes. The sections on set theory, in particular, are a disaster. When he lists the standard axioms of set theory from which mathematicians derive theorems about the iterative conception of a set, he gets the very first one wrong. (It is not, as Wallace says, that if two sets have the same members, then they are the same size. It is that two sets never do have the same members.) From there it is pretty much downhill….
(Via Return of the Reluctant.)