This book is a valuable historical exposition of the development of axiomatic set theory, but it is no longer useful as a textbook, because of the antiquated language. The present book is a Dover 1991 unaltered reprint of the 1968 second edition from North-Holland, which in turn was based on the 1958 first edition. The second edition added a bibliography of more recent works but seems to have no other changes.

Paul Bernays (1888–1977) was one of the pioneers of axiomatic set theory (he also worked in mathematical logic and foundations), and this book represented his take on how set theory should be developed. The book includes a valuable historical introduction to set theory (33 pages) by Abraham Fraenkel.

The axioms presented here are not the widely-used Zermelo–Fraenkel (ZF) axioms but the closely related von Neumann–Bernays–Gödel (NBG) axioms. NBG differs from ZF primarily in allowing classes as well as sets. This was the first book-length exposition of the NBG theory. Most of the book deals not with axiomatics per se but with the elements of set theory, in particular ordinals, cardinals, and transfinite recursion. The coverage is reasonable for an introductory book, although it necessarily omits newer topics such as forcing, filters, and ultrafilters.

The book is hard to follow because the notation has changed a lot since it was written (happily there is a thorough index of notation in the back) and because there are a lot of untranslated German terms, ranging from Aussonderungstheorem to Zahlenklasse.

Although this is not a bad book, time has passed and there are better books today. An oldie but goodie, that’s actually the same age, is Patrick Suppes’s *Axiomatic Set Theory* (Dover, 1972 reprint of the Van Nostrand 1960 edition). It has a much clearer exposition, and is a true textbook with lots of exercises. A more modern book, that I have not seen but that is well-regarded, is Hrbacek & Jech’s *Introduction to Set Theory*. It is a ZF-axiomatic development, but the axioms are introduced as needed and are scattered through the book, with a recapitulation at the end. If you are interested in the machinery of set theory without the axiomatics, Halmos’s *Naïve Set Theory* is a good start.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.