Set Theory (Studies in Logic: Mathematical Logic and Foundations)

INITIAL REVIEWThis will be a short initial review prior to reading any of this very recently acquired book. This book by master expositor Kenneth Kunen, emeritus at University of Wisconsin-Madison, is a newly rewritten 2011 update of his well regarded, rather standard 1980 edition, still available at  Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics) . In Professor Kunen's short new preface, he cites two reasons why the new edition was needed, 1) more set theory has been discovered in the past 30 years, especially related to Martin's Axiom. 2) model theoretic methods are far more more known and used than in 1980. Another plus of this new edition is that it is published in a decent quality, but inexpensive College Publications paperback imprint. I've been handling this book a lot for over a month so far and the paper cover is still flat and undamaged.Long chapter I called 'Background Material' is rather similar to great chapter I on ZFC set theory in Kunen's excellent 2009 book  The Foundations of Mathematics (Logic S.) , which I have read thru 100 page chapter II on model theory and proof theory, with chapter II twice, and finally read short chapter III on philosophy of math. There is a huge review from this reader on the page for that foundations book.BASIC BOOK CONTENTS: Preface-v / 0 Introduction-1 / I Background Material-5 / II Easy Consistency Proofs-106 / III Infinitary Combinatorics-153 / IV Forcing-243 / V Iterated Forcing-314 / Bibliography-388 / Indices-395 (both symbols and subjects) // As in the foundations book, the indexes in this book are really tiny and inadequate. This is again mitigated by Kunen's very effective constant cross-referencing ahead, but mostly behind, and I've found the back-references to be very helpful to my reading. Looking back at contents of the previous edition, which I do not own, that WAS quite a different book than this new one is.READING IN THIS BOOKCHAPTER I / GENERAL COMMENTSRead chapter 0 and started long chapter I on Tue 31Jul12. One thing that strikes me in both of these CP books by Ken Kunen is how involving and insightful he is as a writer. There tends to be a big motivation to keep reading these books. Finished great chap I section 6 on relations and functions on Tue 7Aug12 afternoon, so onward to ordinals next. One improvement in chapter I of this set theory book over chapter I in Kunen's foundations book is perhaps a bit clearer version of getting right into business in the present read, making it a 'smoother' read. Another difference in the chapter 1s is that in this book, he covers a few more subjects than in the other one, so there is less coverage of general set theoretical subjects.Finished both very good sections on ordinals Thu 9Aug12 afternoon. Next section I.9 on 'Induction and Recursion and Foundation' is a large 22 pages long, so that should be a good, long read. Section 9 is appearing to be just way full of Kunen's best wise insights, a real joy to read! One thing that seems to apply mainly in chapter I of this book is that many proofs are skipped, or short proofs are written more in a suggestive way, as if that proof, in an unspoken way, is really the student's job. This bad practice continues in chapter II of this book. For comparison, chapter I of the foundations book contains more thorough proofs of many of the missing or cursory proofs in the present book. Therefore, ideally, a reader should get both of these inexpensive books by professor Kunen, as they are mostly complementary. For example 100 page chapter II of foundations gives an excellent large hit of model theory, which is only touched on in late chapter I of the present book, but model theory is used extensively later in this book.Excellent nuggets of wisdom continue. Finally finished long section I.9 on Mon 20Aug12 and soon read short and interesting I.10 and I.11 on cardinalities. I.12 on the Axiom of Choice and its equivalent statements was especially interesting, ending on p. 72 and finished on Fri 24Aug12. Long I.13 on cardinal arithmetic is nearly the same as I.13 of the foundations book. Tedious going thru this difficult section a second time. Finished I.13, read short I.14 on identity of mathematical objects, and started moderately long I.15 on model theory Mon 27Aug12, going to p. 84. Section I.16 on models for set theory looks forward quite a bit to the work in rapidly upcoming chapter II. Finished I.16, very short I.17 on recursion theory, and whole chapter I Wed 5Sep12 mid day.CHAPTER IIRead thru somewhat informal and short section II.1 on Thu 6Sep12. Longer section II.2 and shorter II.3 will focus on the Foundation Axiom and ends up formally adopting that little-used axiom into this book's WF (well-founded) set theory. That section II.2 is quite interesting and well done. Long section II.4 on 'More on Absoluteness' is some excellent mathematics. The concept of absoluteness was just introduced in section I.16, so here we fill out that theory. Section II.5 on reflection theorems seems to work toward making an end run around some limitations of smaller versions than full ZFC set theory.Following II.5, rather long II.6 on constructible sets was the first one that was brutally difficult for this reader. Murderously difficult definitions and generally hard subjects and symbolism. Skipped most of it, and stopped reading all proofs and exercises/hints during that section. Taking a break from reading this book effective 14:21 Wed 19Sep12. Finished sections II.7 thru II.10 minus exercises, finishing chapter II with my II.6 omissions, etc. on Mon 24Sep12.II.7 was short and vaguely about forcing, while II.8 was about ordinal definable sets and was quite interesting. Then II.9 was strangely about the independence of foundation, which has no effect at all on mathematics. II.10 briefly mentioned set theory with classes as NBG and MK set theories. Kunen has been using proper classes a lot in this book as abbreviations for more complicated logical sentences. ZFC set theory actually contains no proper classes or a universal set, which are strongly used in NBG class-set theory. Here is a book on NBG I have read thru chapter 4:  Set Theory and the Continuum Problem (Dover Books on Mathematics) . Used to have a review of that book for over a year, but deleted it in Aug12, as it seemed to have gone irretrievably bad. In general, chapters in that 1996 Smullyan/Fitting text are awfully short.CHAPTER IIIThis whole chapter on infinitary combinatorics, including Martin's Axiom, is quite difficult to extremely so. At beginning of that chapter, Kunen specifies a total of about 18 of the 90 pages of chapter III for future use, so I will attempt to read that 20% of the chapter before going on to chapter IV on forcing. In total, chapter III contains 8 sections, but recommended reading is in only sections III.1 thru III.3.Started the read in section III.1 Mon 24Sep12 afternoon. The few recommended pages of III.1 feature some really interesting new concepts in cardinals with such cool names as 'meager'/'null'/'non'/'add'/'mad family'. Fascinating section. The only 3 pages of section III.2 deal with topological spaces, and so that is not in my math zone.Section III.3 is a huge 24page section on crucial 'Martin's Axiom' or MA, of which only the first 7 pages are needed. Started that early afternoon on Wed 26Sep12. This is about where things of chapter III start getting more highly difficult. After this chapter III reading, which was finished minus all proofs on Thu 27Sep12, a long time before I just might attempt chapter IV of this book.CHAPTER IVRead good section IV.1, starting chapter IV on forcing mid day on Wed 10Oct12, so we'll see how this chapter goes while reading no proofs or exercises. Gave up continuing in this book after that section IV.1. After chapter II, professor Kunen started writing out much more thorough proofs loaded with more than the usual syntax language symbolic content. That is what makes chapters III and IV appear so very mathematically dense, and for this retired metals guy, dauntingly difficult. Plus, skipping those proofs isn't much of an option, since those proofs are most of the content of these chapters. So this book goes back on my bookshelf. Bye everyone.

Set theory was one of my favorite areas covered in my Introduction to Advanced Mathematics class. I love the subject and this book does a good job presenting the of the foundations of the area. I can spend all day with this book, I get so much joy from it. It gets its elegance from its simplicity.

Good

This book is full of typographical mistakes in the very important places. Figuring out the typos consumes too much time.

Just started it and it looks like the best ever written; way beyond anything previously written. So far:It is flawless in both its delivery and coverage. It is consistent in style and detail, correct in order of events, and complete in coverage. Added is that it reads in a way that will intrigue you if you are slightly familiar with the ideas – you will not be able to put it down. It is clever; Kunen knows not only the subject so well, but also how to present it.The book starts with explaining how language and philosophy is a major concern, and lays down the axioms of various set theories in a provocative way. Kunen takes the important variants of ZFC and shows which axioms are necessary for working in them; in doing so, you see how the axioms fit together to form all of ZFC and any of its cousins. This section is pedantic and must have been painstakingly written, but it reads like a novel and the clarity is quite remarkable. It conveys his in depth understanding, and any willing person can follow it, and with some thought pick up this understanding. There is a back and forth reading that most will take as many of the early ideas are talked about in terms of why and how they will affect the whole theory – so parts of the later exposition is referenced early on. This gives those somewhat familiar with set theory the chance to skim through the whole of the first chapter and then return to get the full punch of the underpinnings. However, this might be a stumbling block for the near novice, but mountable by references given – taking one to the exact definitions of the terms. There is also a handy 2 page symbol reference at the end of the book.Outstanding is the author's constant reminder of the differences of speaking within the meta-theory and the formal system, neither of which are set forth center-stage - they are left to the reader's surmise. This works nicely for those that have prior dealings with both and can visualize actual systems, while for those without this background it really makes little difference for understanding the presentation! Thus a comparatively fluid flow, a canter, is established, devoid of choking formality.Comforting is that a healthy amount of examples are given, especially after definitions. As well, exercises are interspersed throughout the text. I have seen, on the web, starts at the production of answer documents and time will tell if any find their way mainstream and become easily accessible. Chances are they will. The serious reader may want to participate in a collective construction.

I used the first edition from 1980 for teaching forcing to third year and more advanced students.Now Professor Kunen wrote a evenclearer and more versatile exposition. Also the beginnings of proper forcing are presented.It requires time and energy to study the book, that is true. The successful readers will master abeautiful and powerful technique for proving that certain statements do not follow from theZermelo Fraenkel axioms together with Choice if the latter is consistent.

Great book to start set theory after one has seen an introductory course on the basics.Also the price is very low for a math textbook

Not very expensive but very good and interesting , totaly worth it