Here’s one of those science-based textbook reviews that will pop up once in a while.

General Information

Title: Axiomatic Set Theory

Author: Patrick Suppes

Publication Date: 1972 edition of a text first published in 1960

ISBN: 0-486-61630-4

Cover price: $8.95 US

Publisher: Dover Publications

Intended Audience: Late undergraduate, or early graduate audience.

Subject Matter

Set theory covers the most basic aspects of mathematics, providing a formal definition of numbers, addition, and multiplication, among other things. Suppes makes it as easy as {0}, {0, {0}}, {0, {0}, {0, {0}}}.

The breakdown of the chapters is as follows:

  1. Introduction
  2. General Developments
  3. Relations and Functions
  4. Equipollence, Finite Sets, and Cardinal Numbers
  5. Finite Ordinals and Denumerable Sets
  6. Rational Numbers and Real Numbers
  7. Transfinite Induction and Ordinal Arithmetic
  8. The Axiom of Choice

High Point

The fifth chapter was very interesting. It’s amazing to see how numbers are formally defined, and how that definition is actually useful.

Low Point

The introductory chapter didn’t make much sense until after I’d read the next chapter. In general, this isn’t good for a textbook.

The Scores

The clarity of the last seven chapters was very good. As a pretty formal textbook, there’s a lot of symbols that you must wade through, but Suppes is good at restating things in more conventional terms for the sake of the reader when the constructions get complicated. Even though the first chapter had its rough spots, it had some excellent points as well. I give it 5 out of 6.

The structure is very well laid out. The content of the text makes frequent references to the theorems of the past, not just by their numbers, but by the section they appeared in. References to future results that depend on present or coming results are also well marked, saying which results are coming, and why they have been postponed. My only concern is with the Axiom of Choice. The argument for handling it separately given its controversial nature is sound, but I’m concerned by the fact that some results that depend upon it were used several chapters earlier. The text at the time marked them as depending upon the Axiom of Choice, but it’s still not entirely clear to me in retrospect which ones were which. Had those results been postponed until the Axiom of Choice was available, then I’d have a much better mental image of what was going on. I give it 4 out of 6.

The examples were much too rare. Some of the simple ones were provided, but there were so many sections that had no examples of applications that I wondered what the point was. Once functions and relations are established, there should be a way to show any of these given theorems in action, or at least note that a theorem’s only purpose is to prove another theorem. Chapters 5 and 6 were good in this category, but the rest had problems. I give it 3 out of 6.

The exercises were often relevant. Many of them were of a “prove theorem X” nature, which is a good type of exercise for this type of text. Some of the earlier exercises had several “prove or provide a counterexample” exercises of some very touchy and subtle concepts. Without some indication of whether or not the theorem is true, I found it hard to know if my solutions were correct. Essentially, we have no solutions, so the only time I was confident that my solution was correct was when I was so confident in the concept that I didn’t feel I needed a practice exercise. I give it 4 out of 6.

The completeness of the text is good. There’s a full course worth of information here, and the pointers to references that can take you further are frequent and clear. It’s not comprehensive, but it makes note of its shortcomings, and chooses them well. I give it 4 out of 6.

The editing was well done. There are a couple of typos that I noticed in chapter six, but if you’re still reading when you get that far, they’ll be obvious and easy to mentally correct. I give it 5 out of 6.

Overall, it’s a very good introductory book for someone who has already had exposure to formal mathematics. I give it 5 out of 6.

In total, Axiomatic Set Theory receives 30 out of 42.