Book Review: Naïve Set Theory (MIRI course list)


33


So8res

I'm reviewing the books on the MIRI course list. I followed Category Theory with Naïve Set Theory, by Paul R. Halmos.

Naïve Set Theory

Book cover

This book is tiny, containing about 100 pages. It's quite dense, but it's not a difficult read. I'll review the content before giving my impressions.

Chapter List

  1. The Axiom of Extension
  2. The Axiom of Specification
  3. Unordered Pairs
  4. Unions and Intersections
  5. Complements and Powers
  6. Ordered Pairs
  7. Relations
  8. Functions
  9. Families
  10. Inverses and Composites
  11. Numbers
  12. The Peano Axioms
  13. Arithmetic
  14. Order
  15. The Axiom of Choice
  16. Zorn's Lemma
  17. Well Ordering
  18. Transfinite Recursion
  19. Ordinal Numbers
  20. Sets of Ordinal Numbers
  21. Ordinal Arithmetic
  22. The Schröder—Bernstein Theorem
  23. Countable Sets
  24. Cardinal Arithmetic
  25. Cardinal Numbers

Normally I'd summarize each chapter, but chapters were about four tiny pages each and the content is mostly described by the chapter name. Zorn's Lemma states that if all chains in a set have an upper bound, then the set has a maximal element. (This follows from the axiom of choice.) The Schröder-Bernstein Theorem states that if X is equivalent to a subset of Y, and Y is equivalent to a subset of X, then X and Y are equivalent. The other chapter titles are self-evident.

Each chapter presented the concepts in a concise manner, then worked through a few of the implications (with proofs), then provided a few short exercises.

None of the concepts within were particularly surprising, but it was good to play with them first-hand. Most useful was interacting with ordinal and cardinal numbers. It was nice to examine the actual structure of each type of number (in set theory) and deepen my previously-superficial knowledge of the distinction.

Discussion

Before diving in to the review it's important to remember that the usefulness of a math textbook is heavily dependent upon your math background. I have a moderately strong background. Some specific subjects (analysis, type theory, group theory, etc.) have given me a solid, if indirect, foundation in set theory. This was the first time I studied set theory directly, but the concepts were hardly new.

Overview

I was pleased with this book. It is terse. It has exercises. It gives you information and gets out of your way, which is what I like in a textbook: It doesn't waste your time. I'm about to harp on the book for a spell, but please keep in mind that my overall feeling was positive.

Please take these reviews with a grain of salt, as sample size is 1 and I have not read any similar textbooks.

Complaints

  • The book was written in 1960, and it shows. Set theory is more mature now than it was then. The authors often remark on syntax that was not yet standard (which is now commonplace). The continuum hypothesis had not yet been proven unprovable in ZFC. The axiom of choice is embraced wholeheartedly with no discussion of weaker variants. The style of proof differs from the modern style. None of this is bad, per se. In fact, it's quite a fascinating time capsule: I enjoyed seeing a slice of mathematics from half a century past. However, I believe a more modern introduction to set theory could have taught me more pertinent mathematics in the same amount of time.

  • The notation is inconsistent. I've long believed that math is a poor and inconsistent language. This is evident throughout set theory. To the author's credit, they point out many of the inconsistencies: f(A) can refer to both a function or a restriction of a function to the subset A of its domain, 2^w can refer to either functions mapping w onto 2 or a specific ordinal number, etc. I am personally of the opinion that introductory textbooks should enforce a pure & consistent syntax (which may be relaxed in practice). I was mildly annoyed with how the authors acknowledged the inconsistencies and then embraced them, thereby perpetuating a memetic tragedy of the commons. (I know that I shouldn't expect better, but one can dream.)

  • The proofs given were primarily in english. Not once did the authors write ∃ or ∀. They would resort to "for some" or "for any" in largely english-language proofs. The proofs were rigorous (the authors tightly restricted their english phrases), but I was somewhat surprised to find the axioms of set theory described in lingual (rather than symbolic) form.

  • Set theory is axiom soup. I do not view set theory as foundational. Is the axiom of choice true? The question is poorly formed. Axioms are tools to constrain what you're talking about. Better questions are shaped like "does the axiom of choice apply to this thing I'm working with?", or "how does the structure change if we take this statement as an axiom?". This sentiment seems fairly common in modern mathematics, but it was lacking in Naïve Set Theory. Axioms were presented as facts, not tools. There was little exploration of each axiom, what it cost and what it bought, and what alternate forms are available.

Most of these gripes are small compared to the amount of good data in the book. Remember that the book is titled Naïve Set Theory: a little naïvety is to be expected. The takeaway is that the book was good, but likely could have been better in light of modern mathematics. All in all, the book covers lot of ground at a fast clip, and was quite useful.

Should I learn set theory?

As always, it depends upon your goals. Set theory is everywhere in mathematics, and I personally appreciated shoring up my foundations. If you have similar goals, you can easily go through this book in a week if you think that learning set theory is worth your time.

I don't particularly recommend set theory to armchair mathematicians. In my experience, other areas of mathematics are much more fun from a casual standpoint. (Group theory and information theory come to mind, if you're looking for a good time.)

Should I read this book?

Maybe. I have no point of comparison here. My tentative suggestion is that you should find a more modern (but similarly terse) introductory textbook and read that instead. (If you have a good suggestion, you should leave it in the comments.)

I found this book to be rather basic. If you have a background similar to mine, I recommend something a little more advanced. (Unfortunately, I can make no recommendations. Again, comments are welcome.)

This book seems well-suited for a layperson interested in learning set theory. The 1960s feel is definitely fun. I would guess that the book is well-paced for someone who has done the standard college calculus courses but is unfamiliar with Set Theory subject matter.

What should I read?

If you're going to read the book then I suggest reading the whole thing. It builds from first principles up to cardinality, and nothing along the way is unimportant. My only suggestion is that you swap chapter 25 and 24: they appear to have been ordered incorrectly for political reasons. (The derivation of cardinal numbers used in chapter 25 was, at the time, controversial, so the book presents cardinal arithmetic before cardinal numbers.) Other than that, the book was well structured.

Final Notes

If a comparably short-and-sweet textbook written in the last twenty years can be found, I recommend updating the suggestion on the MIRI course list. It's not clear to me how much raw set theory is useful in modern AI research; my wild guess is that mathematical logic, model theory, and provability theory are more important. If that is the case, then I think the technical level of this book is appropriate for the course list: it's sufficient to brush up on the basics, but it doesn't send you deep into rabbit holes when there are more interesting topics on the horizon.

My next review will take more time than did the previous four. I have a number of loose ends to tie up before jumping in to Model Theory, and I have much less familiarity with the subject matter.