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! 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
There are some things that probably should be questioned about long chapter II. 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.
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.
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.
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.
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.
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.
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. (Not quite)
CHAPTER IV / SEP 2013
On Wed 11Sep13, I decided to read chapter IV of this book after all, and got started on that reading. I think this chapter will be great after recently reading into chapter 8 of Theory of Recursive Functions and Effective Computability, which is heavy duty recursion theory. Kunen's version of recursion theory is tied up firmly with the set theory of this book. As a matter of fact it is almost just a further treatment of his 'HF' Hereditarily Finite set theory, introduced in chapter I, along with models of set theory, which has me going back to read later portions of great 13pg section II.17, which is frequently referenced in chapter IV. Back to chapter IV on Tue 24Sep13. Some excellent insightful discussion within long section IV.3. Section IV.4 on diagonal arguments was also quite interestingly written, finished on lovely Sun 29Sep13 afternoon. Only section IV.5 on decidability in logic remains to finish reading this book. It has to be said that it seems to me that Kunen badly mis-titled chapter IV as 'Recursion Theory'. As I've been writing, this is more HF set theory and bits of model theory under another and incorrect name. Finished chapter IV and this book in early Oct 2013, after 2 years of intermittently reading and rereading it. Thru all this reading of chapter IV, it has taken me away from my main present read, the great The Lambda Calculus. Its Syntax and Semantics. Fully back to that now!