Book note: Kunen, Foundations of Mathematics

Finally, here’s the last book in my ‘must-revisit’ stack! — Kenneth Kunen’s The Foundations of Mathematics (College Publications, 2009). Now, I’m going to avert my gaze from some of the philosophical asides here. Kunen writes, for example,

Presumably, you know that set theory is important. You may not know that set theory is all-important. That is:

All abstract mathematical concepts are set-theoretic.All concrete mathematical objects are specific sets.

Abstract concepts all reduce to set theory.

Really? Really? … Well, fortunately you don’t at all need to buy into such obiter dicta (or into the brief philosophical Ch. III) to find some of Kunen’s technical expositions interesting and helpful.

Ch. 1 (77 pp.) is on set theory, shaped by presenting the axioms of ZFC, unfolding their content and significance, getting as far as talking about ordinals and cardinals, choice, the role of the axiom of foundation, etc. This is clearly done: the chapter could suit mathematicians already a little familiar with sets-in-use from their algebra or topology courses, and/or will make a sharp and useful follow-up — one step more sophisticated but still relatively elementary — if tackled after an entry-level set-theory introduction like Enderton’s.

Ch. 2 (100 pp.) discusses some model theory and proof theory. But, in headline terms, I really didn’t find this chapter as accessible and helpful as the previous one.

Ch. 4 (50 pp.) is on recursion theory. And here a key link is made with the first chapter by construing the inputs and outputs of computable functions as hereditarily finite sets. This is a neat device that puts us in the neck of the woods explored by Melvin Fitting’s lovely book Incompleteness in the Land of Sets. And in fact, you’ll probably get much more out of tackling interesting Kunen’s chapter by reading Fitting first (as well as a more conventional introduction to recursion theory).

In briefest terms, the first and last chapter are, I think, recommendable.

And here I’ll pause the flurry of book notes on some relatively introductory logic texts. As I said before, these have really been written as aides memoires for myself, as I start thinking about the next edition of the Beginning Mathematical Logic Study Guide. But I find that if I make the effort to turn telegraphic jottings into posts here, it does help considerably to concentrate the mind and fix ideas. So make of these recent posts what you will!

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