The book has two main problems. The first is Goldstein's explication of the incompleteness theorem. The theorem is the reason for reading a book about Goedel. For the most part, the worth of a book about him for a general reader is measured by the clarity of an explication of the theorem. Goldstein's audience comprises readers who are not logicians or mathematicians, and so a lack of rigor is expected (p. 172). But Goldstein simplifies too much. Her explication is somewhat less clear than both the longer explication in Goedel's Proof by Nagel and Newman and the more technical introduction by Braithwaite in the Dover Publications reprint of Goedel's original paper.
Goldstein's numbering system (p. 172-175) is an example of oversimplification. Goedel's numbering system "used the exponential products of prime numbers and relied on the prime factorization theorem which states that every number can be uniquely factored into the products of primes" (p. 172). In this way, the "metasyntactic relation of provability will become an arithmetical relationship" (p. 176). Under Goldstein's simplified system of numbering, however, it is not at all clear that provability relationships among propositions would translate into arithmetical relationships among numbers, as they do under Goedel's numbering system. Why does Goldstein offer an alternative numbering system for illustrative purposes? I can't tell. She says that her system, if it were made rigorous, would be just as complicated as Goedel's own system (p. 172). So rather than invent her own, why doesn't she just set out a non-rigorous version of Goedel's own system? (That is what Nagel and Newman do.) Not only does Goldstein not improve matters, but also she loses clarity. In her illustration, for example, she says: "Suppose that GN(wffsub1) = 195589 and GN(wffsub2) = 317" (p. 175). But under Goldstein's own numbering system, 195589 and 317 would correspond, respectively, to ~'00)' and x~(, neither of which is a wff! By oversimplifying, Goldstein has made a mess.
The second main problem with the book is Goldstein's fascination with Wittgenstein and her comparison of him with Goedel. Any comparison between the two thinkers feels strained to begin with, and Goldstein's book does nothing to allay that feeling. It is a bit like writing a book about Vladimir Horowitz and then devoting a third of the book to comparing him with Liberace. Moreover, apart from whether any comparison is useful, Goldstein refuses to take Goedel at his word when he says that Wittgenstein had no influence on his work (p. 115-116). In fact, Goldstein takes Goedel's emphatic denial, coupled with what she sees as Goedel's resentment of Wittgenstein (p. 89), as evidence that Wittgenstein must have had an influence on Goedel, or "incentive" or "significant, if ambiguous, role," as Goldstein puts it (pp. 89, 116). This is just weird. If Goedel had written that he hated rock candy and didn't even like the looks of it, would Goldstein conclude either that Goedel really did like rock candy or that he ate filet mignon as a substitute? The ink that she spills on Wittgenstein could have been put to better use on Turing or von Neumann, both of whom get too few words.
The book contains strange repetitions. For example, twice Goedel's work is compared to Alice in Wonderland (p. 170, 252) and twice Goldstein tells us that her New York apartment has only one bathroom (p. 140, 184). Her catty remarks about Goedel's wife, his diet, and their home decorating are rude and irrelevant (pp. 208-209, 223). How poorly Goedel dealt with faculty politics is dull and irrelevant (pp. 234-245). Some extra proofreading wouldn't have hurt, either: "GN(p)" should be "GN(psub1)" (p. 174); "tilda" should be "tilde" (p. 174); "swiped" should be "swapped" (p. 210); and Waismann's name is misspelled twice (p. 105).
The book has its good points. The stories of Goedel's quirks and his friendship with Einstein are entertaining, the sketch of the Vienna Circle is okay, and Goldstein is right to point out that Einstein and Goedel should not be lumped together with Bohr, Heisenberg, and others as "destroyers of objectivity" (pp. 38-39). But that's about as good as the book gets.