The Foundations of Mathematics (Logic) [Englisch] [Taschenbuch]

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)

GENERAL INFO
This 260 page book by Kenneth Kunen, emeritus at the University of Wisconsin, Madison is a good, low cost textbook published in 2009. There are only four main chapters, three of them quite long. A short chapter 0 Introduction starts things off, but the real chapters are I Set Theory (ZFC) / II Model Theory and Proof Theory / III The Philosophy of Mathematics (the short one) / IV Recursion Theory. The many exercises are scattered throughout the book and some are not phrased so that the student even knows what is being asked for.

LAST THINGS FIRST
The final subject index in this book is tiny and not at all detailed. However, Kunen's talent for clear, insightful prose, and his frequent cross referencing both forward and backward within the book, plus the detailed table of contents make the tiny index mostly forgivable in my opinion.

SUGGESTED REQUIREMENTS OF READERS
This book is somewhere in the spectrum of graduate level, and knowing some previous ZF set theory, including the alephs, much general mathematical logic, and a fair amount of abstract algebra would be good preparation before reading this text. Be warned that if you plan to work exercises, many of them range widely across mathematics.

CHAPTER I ON ZFC SET THEORY
Chapter 1 on set theory is about 80 pages long and covers all the nine ZFC axioms for starters, along with other 'smaller' set theories depending on whether Choice (Axiom 9), Replacement (Axiom 6), or Foundation (Axiom 2) are accepted or not. The most restrictive set theory is known as Z-, in which all three of those axioms are rejected. After discussion of 4 of the axioms in some detail, there is a long, very good section on 'Relations, Functions, and Discrete Mathematics', followed by two detailed sections on the ordinals, including ordinal arithmetic. Then a short section on power sets, and a detailed first section on the cardinals, which I had no idea are actually a certain kind of ordinals! Next is a section on the axiom of choice, followed by information on cardinal arithmetic. The large study of the axiom of foundation points out how little that axiom is actually used/needed. Final section 15 of this chapter on set theory is about real numbers and symbolic entities. Chapter I ends on p. 85. This reader strongly enjoyed reading chapter I.

CHAPTER II ON MODEL THEORY AND PROOF THEORY--READ AGAIN IN MAY 2012-SEE FAR BELOW
By 1 October, I was reading in middle of 100 page chapter II on model theory and proof theory, which contains 19 subject sections. After some background sections, Kunen unusually in section 4 explains and adopts the strictly prefix Polish Notation (Jan Lucasiewicz-1920s) as the official formal system for studying model theory. For this reader postfix Reverse Polish Notation (RPN) seems a lot easier to grasp than prefix PN due to his several Hewlett Packard RPN calculators. Then in section 6 on abbreviations, Kunen realizes that PN is "painful" and mostly adopts the less formal normal first-order logic syntax and semantics. There is a whole lot of coverage of semantics thru section 8, and I finally skipped the last three pages of it. By section 10 Kunen is introducing the first methods of proof in the chapter, based on 11 axioms and modus ponens. Section 11 covers proof techniques, which I do find tedious, and then long sections 12 and 13 are about the completeness theorem and other complete theories, respectively. Section 12 proves 13 often very long lemmas on a variety of newly defined objects to make the final completeness theorem proof very easy. That long, complex sequence is absolute torture for this reader! I am starting to question anything that Kunen wrote in this 100 page chapter II. Heavy duty model theory follows the completeness sections, including models for ZF set theory and both the upward and downward Lowenheim-Skolem theorems, which were stated in an incomprehensible way. Weaker set theories and info on other set theories conclude chaper II. Chapter II seems to go from first year graduate to a much higher academic level over its course, and ends on p. 185.

CHAPTER II CONTINUED / END OF READING
At start of section section 11 on methods of proof (p.122), this book for me seems to become suddenly much more difficult. On p. 135 in the miserable section 12 on completeness, I stopped reading all proofs, so I finally stopped reading this book in section 15 on p. 149 and Thu 6Oct11, and had been reading this book steadily and informally since Thu 1Sep11.

QUESTIONING CHAPTER II
As far as questioning the entire huge chapter II of this book, not only was section 12 interminable with its over a dozen lemmas and their proofs, but why on earth did Kunen thoroughly introduce Polish Notation in section 4 as THE language for model theory, only to scrap it in section 6 for normal first-order logic? Then the impenetrable statements of theorems in section 16 make me wonder if anyone could understand what they even say. Therefore, I think the author's expected sense of good judgment was totally missing in that chapter!

CHAPTER III ON PHILOSOPHY OF MATHEMATICS
Very short chapter III on philosophy of math is only about 8.5 pages long and is still fairly technical, but much easier than everything surrounding it. Vacation!

CHAPTER IV ON RECURSION THEORY
Chapter IV on Recursion Theory, at about 50 pages, seems to be at a level of difficulty similar to late chapter II, i.e., quite difficult for a mathematical layperson such as me. It also takes an unusual set-theoretic approach to recursion theory. Nevertheless, I've read much on the theory of computation, including all manor of recursive sets and functions, so chap IV of this book was not much on my plate to read anyway.
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REREADING CHAPTER II--MAY 2012
In May12, I plan to reread difficult chapter II on model theory and proof theory, but this time not getting bogged down in all those proofs. I plan to just read the descriptive prose, the definitions, and the lemmas and theorems, while reading NO PROOFS in the whole chapter. This will be in a second copy of this book.

SECTIONS 1-9
Started the rereading on Tue 1May12 afternoon. In the first 9 of 19 sections of this 100 page chapter, it seems like Professor Kunen lacks a strong focus. He tends to wander all over math a lot. For example, in section 5 on syntax of first-order logic, he ends up getting into group theory, which seems tenuously connected to syntax of formal logic. And in section 5, he is still trying to use strictly prefix Polish Notation, which is fine in my Scheme programming language, but is highly difficult for symbolic logic. Short section 6, as I said above starts to give up Polish notation in favor of regular first-order logic (FOL). Sections 7 and 8, in an ad nauseum way do semantics of FOL in huge detail, again wandering thru mathematics in the process. Group theory and field theory from algebra seem to be Kunen's favorites to pull us thru repeatedly, so I am starting to figure out what he is getting at with those maths in this context.

SECTIONS 10-16
For comments on sections 10 and 11 on the proof theory material, see my main chapter II writing from last fall above. Again, proof theory isn't a favorite of mine. A count of all the proofs in long section 12 on the completeness theorem: 16 proofs in 10 pages, all to be skipped this time. Section 12 IS definitely more interesting and palatable when reading all but the proofs. The completeness theorem itself links the syntactic provability of an expression to its semantic, model theoretic aspects, and as such is a big unifier of much in this long chapter. Finished section 12 / started section 13 on complete theories on Sat 12May12. Late section 14 on 'Equational and Horn Theories' starts getting more difficult, while section 15 on 'Extensions by Definitions' eases off a bit and is quite interesting and fairly practical. I still can't figure out what the theorems are telling me in section 16 on elementary submodels, as they are stated in an extremely abstract way, noted by me above in last fall's writing. Finally skipped reading the last several exercises/hints at end of section 16 late day Tue 15May12. I have generally been reading even the exercises and hints in this read.

SECTIONS 17-19
The end of this long chapter is in sight. Long section 17 on subjects around models of set theory is fascinating, and lets me review portions of excellent Chap I on set theory to follow this section. I am really starting to become a fan of model theory due to this chapter. Long, good, but increasingly difficult section 17 was finished and then started amazing section 18 on weaker set theories in pm of Sat 19May12. It is going to be fascinating to figure out what happens if we gut the ZFC axioms and even make a new set theory PAS based on Peano Arithmetic, universally called 'PA'. Should be a great ride! Section 18 is some of the most interesting mathematics I have ever read, truly amazing! Interesting final section 19 mostly on computerized proof methods was finished mid day Mon 21May12, so 3 weeks to finish ALL but the proofs of this now excellent 100 page chapter II.

CHAPTER III
On Mon 21May12, I did read the quite interesting short 'vacation' chapter on philosophy of mathematics. It mostly compared various versions of 'platonist', 'finitist', and 'formalist' approaches to math, focusing on the current topics of set theory, logic and proof theory, and model theory. I still have no intention of reading Kunen's version of recursion theory in chapter IV, so this is the end of reading this very good book for me.