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A Course on Set Theory

Ernest Schimmerling
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
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This thoroughly modern introduction to set theory is aimed at undergraduates with a firm and thorough basis in proof-based mathematics and the usual calculus. Lebesgue measure theory would be helpful, but is not required.

The notation of formal proofs is assumed to be understandable. It is key that the reader can be comfortable with a statement like ∀A ∃!B x [xB ⇔ (x A and P(x))], with really only the symbol ∈ having been formally introduced, in order to even grasp the fundamentals of Chapter 2. This preliminary chapter lays out the axiomatic basis of the presentation of Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC).

Beginning with the axioms of ZFC and then rocketing immediately into a rigorous description of set theory is what marks this volume as thoroughly modern and is one reason it is also compact. Initially, I was put off. Who are these students so adept at formal proofs yet stopped in their tracks by the membership symbol? Setting such scruples aside, I laminated a bookmark with an overview of Russell‘s Paradox and read on. In reading on, it became obvious that this material has been road tested over the years with students at Carnegie Mellon University. The economically presented definitions, theorems and lemmas are supported by well-timed and relevant exercises all having the hallmarks of being tried and tested by the author in his undergraduate classes.

Covering the material in 161 pages is a breathless pace when considering that this includes cardinality, trees, linear orderings, well orderings, filters, and ideals. Readers will be not only pondering Cantor before a third of the book is passed, but need a ready understanding of the taxonomy of transfinite numbers.

The meat of this text, and an indisputable one, is that set theory is the mathematics of infinity. While the book has a modern approach and scope similar to other available texts like Introduction to Modern Set Theory by Judith Roitman, the early introduction of ZFC and the transfinites gives the book its own flavor. For instance, filters and ideals can be introduced with only the cardinality of the infinite set of the natural numbers at hand, especially in an introductory course. This and other topics coming later than may be expected allows them to be employed in further probing what lies beyond ω.

This is an excellent, self-contained outline suitable for a one-semester introductory course. Backed up by lecture, I can see this succeeding very well. I am more concerned for the independent reader looking for an introduction, even assuming a suitable mathematical background. The exercises have no solutions, but there are strategic hints and they tie so well to the relevant text that I see no problem there. I would have two small improvements: first, a more thorough, lengthy index, and second the foundational ZFC axioms should be numbered as an aid for when they are referred to later in the text.

Tom Schulte ponders ideas from aleph to omega and teaches mathematics at Oakland Community College in Michigan.

Note to the instructor
1. Preliminaries
2. ZFC
3. Order
4. Cardinality
5. Trees
6. Dense linear orderings
7. Filters and ideals
Appendix. Summary of exercises on Boolean algebra