The Foundations of Mathematics (Studies in Logic: Mathematical Logic and Foundations)

This is the second book written by Kunen I have read. In his book Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics), he gives a brilliant exposition of the basic techniques to proof statements to be consistent with Zermelo-Fraenkel Set Theory.The reason I bought this book is the same reason I bought the first one: I know nothing about the subject and the book looks like a promising way to learn the basic stuff. I am a topology Ph.D. and have found that it is very common to have many Set Theoretic and Model Theoretic tools in the area I want to specialize in. Further, there is no one who will help me learn this at college, so I decided I had to study at least the basic stuff by myself in order to learn what I needed. I think both of Kunen's books are great to learn by oneself. However, many undergraduates have told me that Kunen's first book is hard to read so you must take caution at my words.One of the most amusing features of Kunen's book is that the way he explains things is really entertaining. This feature is also present in this book in various places, for example: when he explains the notion of Cardinality (p. 17) by talking about ducks and pigs, when he compares the philosophy of mathematics with religion (p. 190) or my favourite quote from the part where he explains the philosophy of the Church-Turing thesis (p. 200):"Likewise, the ultimate nature of human intelligence and insight is not understood, so it is conceivable that a human, via some insight, perhaps in contact with God or the spirit world, could reliably decide membership in some non-$\Delta_1$-set."I must of course make clear that this book is serious and even though this quote seems taken from a pseudoscience book, it has philosophycal roots in the reductionist philosophy he then explains.Now let me give my opinion about each of the chapters.Set Theory: In this chapter, he develops the basic facts about ZFC set theory in a very detailed fashion, he practically gives all the missing details from his first book. He also talks about some simple models like $R(\gamma)$ and $H(\kappa)$ which will play an important role in chapter 2. One of the most interesting exercises in the book is I.14.14 where he gives you a hint on how to construct a computable (not yet defined) enumeration of hereditarily finite sets. An excelent summary on the subject.Proof Theory and Model Theory: This chapter is almost 100 pages long but is worth reading. Even though the details of proof theory seem annoyingly tedious, this is the first book in which I venture to read them all. I was really worth it and I finally feel myself "complete" by knowing what the completeness theorem says and how to prove it. Of course, the most interesting parts (in my opinion) are the ones that come next, that are model theoretic. Elementary submodels were practically the reason I wanted to read the book and I think they were neatly explained. After this, he starts talking about absoluteness in models of set theory, something which is in more detail in his previous book. He also manages to introduce the notion of $\Delta_0$ (which for the first time I understood) and $\Delta_1$, which will be used in chapter 4. A really complete survey.Philosophy: There's not much to say here because the content is purely philosophycal (and really interesting) but non-mathematical, you should read it by yourself.Recursion Theory: This chapter was kind of boring for me. Although I finally came to understanding the Church-Turing thesis: "A subset of HF is computable if and only if it is $\Delta_1$ in HF", the arguments start becoming hard to follow. Many details are missing which made me sometimes despair on what the arguments really meant. By this part of the book, you have learned that many things you take for granted in the general setting are important details in logic, so it feels bad when Kunen starts skipping details. Of course, the author says himself he will skip more and more details as he advances, so it may be my inexperience in this kind of arguments. Important theorems that I have managed somehow to understand (at an informal level) are Godel's incompleteness theorems and Tarski's undefinability of truth. One thing I can say I learned is that the absoluteness of hereditarily finite sets is everything that matters for this theorems (or so I understood). A good excuse for the apparent lack of "Kunenness" of this chapter is that he refers to the book Incompleteness in the Land of Sets (Studies in Logic) for a more extensive reading.The bibliography is extense and not only mathematical. There are links to web pages where you can find ancient texts (like one from Ockham) and computer programs that simulate proof theory. The bibliography is definitely worth reading.One last detail is that I noticed that Kunen does not talk about Category Theory. I have read elsewhere that there are parts of logic that are studied in the categorical point of view. Plus, Kunen says that Set Theory is the "theory of everything". However, there are mathematicians that have done research on trying to axiomatize mathematics using categories. I even remmember there was some kind of (heated) discussion between Mac Lane and Mathias on whether category theory should replace set theory. Of course I am of the idea that set theory is more fun. However, I think Kunen lacks a discussion about this matters. He does mention categories once "in the language of category theory" in some part of the book to make a notion easy to understand (however, I cannot find it anymore) but that is the only time when he talks about this. Thus, I think what this books lacks is a section on categories.Conclusion: you will find this book interesting if you are interested in a "fast" reading on foundations of mathematics, the references should guide you to more advanced topics and specialization of the ideas presented, read only if you are mathematically mature (perhaps for grad school)

It seems like a good book. So far I haven't read a lot of it.

I am not a mathematician, but I've been interested in this topic for a number of years. I have read a variety of books on the topic of foundations of mathematics written for the general public and had an undergraduate course in foundations a million years ago when I was in college, but have never taken the time to read through an actual math text book on the topic. I understand modern algebra from studying Dummitt and Foote on my own a few years ago, and the real numbers from studying Roydon, again on my own. (Somehow working in the real world has always gotten in the way of any serious pursuit of academia for me). Anyway, I present this bit of personal history by way of saying that I found this book to be an awesome text book for me for a formal introduction to set theory (ZFC and otherwise), model theory, and recursion theory with a nice dose of philosophy to spice up the mix. Really great. I kept up with working out the exercises until somewhere around strongly inaccessible cardinals, but not being a mathematician or formal student I have the luxury to pass on doing the hard work and to just read the book if I want, (and yes, I can feel you judging me for saying that).A really great thing about this book is that Kunen gives elucidating hints for the exercises that point the way enough that even for a non-professional like me who doesn't care to go deeply into working out exercises, I still get to have a insight as to how the solutions will work out. I super appreciated that and Kunen is consistent in this approach. The exposition is accessible throughout, so staying focused and disciplined in reading on my part was sufficient for getting through the material.I can say that I would have truly loved to have had an instructor that could have provided some coaching and help with the exercises along the way (in which case I would have been very motivated to try many of the exercises I passed on); which is to say, I believe it would be an excellent text book for a teacher that really wants to provide the coaching part of the learning equation to, say, first year graduate students using this book.

As a working mathematician with no real knowledge of logic beyond the rudiments, I bought this book on a colleague's recommendation. I've been extremely pleased. I spent a couple months working through it cover to cover, and was able to gain a decent feeling for what model, proof and recursion theory are all about, and the connections between them. I worked most of the exercises and found them very helpful and of appropriate difficulty. Kunen's writing is very pleasant to follow, and he includes not only technical but also philosophical background.The book is well edited: I have a good eye for typos and spotted only a handful, most of them simple spelling or punctuation errors that have no impact on the content. The printing and paperback binding are of high quality despite the incredibly low price; it has stood up well to being thumbed, flipped, thrown open on the floor, etc, for many weeks.This would be a great textbook for a first graduate course or an advanced undergraduate; there are really no prerequisites besides mathematical maturity. Reading it made me wish I had taken such a course.My only complaint in retrospect is that I had some trouble distinguishing between arguments working in set theory and in the metatheory; Kunen could have done better at pointing out which is which. This would be less a problem for the book as a textbook, where the instructor could clarify where needed, but for self-study it was occasionally confusing. Nevertheless, I came away extremely satisfied and would recommend this book to anyone looking to better understand mathematics' foundations.

Fine