# Teach yourself logic, #4: Beginning set theory

Here, after rather a long gap, is another instalment in the “Teach Yourself Logic” series. For new readers: given the dire state of logic teaching in some grad schools in philosophy (especially in the UK), I’m trying to put together a helpfully annotated reading list. The aim is to give students needing to teach themselves some logic a Guide through the daunting yards of books that are (or ought to be) in their university library. The reading list might be helpful to some mathematicians too. What we’ve covered up to now falls into four sections:

1. Back to the beginning
2. Getting to grips with first-order logic
3. Modal logic
4. From first-order logic to model theory

And here’s an edited version of the list so far. But I’m next going to jump out of sequence to what is planned to be §9 of the Guide, covering set-theory, since that’s what I happen to be thinking about right now. So now read on …

Where to start?

• Derek Goldrei, Classic Set Theory (Chapman & Hall/CRC 1996) is written by a lecturer at the Open University in the UK and has the subtitle ‘For guided independent study’. It is as you might expect extremely clear, and it is quite attractively written (as set theory books go!).
• Winfried Just and Martin Weese, Discovering Modern Set Theory I: The Basics (American Mathematical Society, 1996). This covers overlapping ground, but perhaps more zestfully and with a little more discussion of conceptually interesting issues, though also it is at some places more challenging (the pace can be uneven). But this is evidently written by enthusiastic teachers, and the book is engaging.

My next suggestion some might find a bit surprising, as it is a blast from the past. However, both philosophers and mathematicians ought to appreciate the way it puts the development of our canonical ZFC set theory into some context, and also discusses alternative approaches:

• Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, Foundations of Set-Theory (North-Holland, 2nd edition 1973). This really is attractively readable, and should be very largely accessible at this early stage. I’m not an enthusiast for history for history’s sake: but it really is worth knowing the stories that unfold here.

One intriguing feature of that last book is that it nowhere mentions the idea of the ‘cumulative hierarchy’ — the picture of the universe of sets as built up in a hierarchy of  levels, each level containing all the sets at previous levels plus new ones (so the levels are cumulative). This picture — nowadays familiar to every beginner — comes to the foreground in

• Michael Potter, Set Theory and Its Philosophy (OUP, 2004). For philosophers (and for mathematicians concerned with foundational issues) this is — at some stage —  a ‘must read’, a unique blend of mathematical exposition and conceptual commentary.  Potter is presenting not straight ZFC but a very attractive variant due to Dana Scott whose axioms more directly encapsulate the idea of the cumulative hierarchy of sets. However, it has to be said that there are passages which are pretty hard going — sometimes because of the philosophical ideas involved, but sometimes because of unnecessary expositional compression. In particular, at the key point at p.  41 where a trick is used to avoid treating the notion of a level (i.e. a level in the hierarchy) as a primitive, the definitions are presented too quickly, and I know that real beginners can get lost. However, if  you have read Just and Weese in particular, you should be able to work out what is going on and read on past this stumbling block.

The books mentioned so far don’t mention but don’t treat (1) independence and consistency results, and though Potter mentions (2) large cardinals, again this is without any development. For a first look at (1), a possibility is

• Keith Devlin The Joy of Sets (Springer, 2nd end. 1993). This is again well written, but goes significantly faster than Goldrei or Just/Weese,  Chs 1–3 giving a fast track coverage of some of the material in those books. Later chapters in this compact book introduce more advanced material. In particular, Ch. 5 discusses Gödel’s notion of constructible sets, Ch. 6 uses “Boolean valued” sets to prove the independence of the Continuum Hypothesis, and Ch. 7 considers what happens if you allow non-well-founded sets (infinite downward membership chains). But all this is done pretty speedily, which may or may not appeal.

But you could well jump over Devlin and go straight to a more comprehensive treatment of independence proofs, the self-selecting

• Kenneth Kunen Set Theory: An Introduction to Independence Proofs (North Holland, 1980), rewritten as his Set Theory* (College Publications, 2011). The first version is a modern classic, used in many university courses: the new version is a timely update and — on a fairly brief inspection — looks to have all the virtues of the earlier version, but acknowledging thirty years of progress.

And then of course — if you’ve got the set-theoretic bug — the modern bible awaits you in the form of the rather monumental

• Thomas Jech, Set Theory: The Third Millenium Edition (Springer, 2003).

Which is more than enough to be getting on with!

However, as an afterthought in small print for the list, we might usefully add a handful of other books that seem to me of particular interest for one reason or another. These could all be read after Goldrei or Just/Weese before or alongside Kunen. In no special order

• Thomas Forster, Set Theory with a Universal Set: Exploring an Untyped Universe (OUP, 2nd end. 1995). Focuses on Quine’s NF and related systems. It is worth knowing something about alternatives to ZFC.
• Thomas Jech, The Axiom of Choice* (North-Holland, 1973: reprinted by Dover Books 2008). Readable, attractively short, and will tell you about a variety of constructions (including Fraenkel-Mostowski models and Cohen forcing).
• Raymond Smullyan and Melvin Fitting, Set Theory and the Continuum Problem*  (OUP 1996, revised edition by Dover Books 2010). Famously lucid authors trying to make hard proofs accessible and doing a good job.
• If you were intrigued by some of the historical material in Fraenkel/Bar-Hillel/Levy then you should enjoy José Ferreirós, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics (Birkäuser, 2007). This is slightly mistitled — it is the history of early set theory, stopping around Gödel’s relative consistency results. But a very interesting read.

I could go on! Should I have mentioned Moschovakis’s nice Notes on Set Theory earlier on in the main list? Added Bell on Boolean-Valued Models to the small print?? Even added Halbeisen’s interesting new book on Combinatorial Set Theory??? But let me pause here.

Added later I surely should have mentioned at the outset Enderton’s book Elements of Set Theory (what a pity it isn’t available as a cheap Dover reprint!). And I am beginning to wonder about splitting this burgeoning section of the Guide into two sections — ‘Beginning Set Theory’ and ‘Continuing Set Theory”.

### 12 thoughts on “Teach yourself logic, #4: Beginning set theory”

1. There are two other introductory level books that might be worth mentioning. First is Hrbacek and Jech’s Introduction to Set Theory. I am not sure how it compares with the other intro books on the list. Second is Schimmerling’s A Course on Set Theory. It is an intro book that is geared more towards some of the computational issues that arise in Moschovakis’s book. I would also support including Moschovakis’s Notes on Set Theory in the list. It is very nicely written. (Perhaps it is not appropriate to include in this list, but Moschovakis’s Descriptive Set Theory should go on some list.)

Barwise’s Admissible Sets and Structures has a clear introduction to the set theory KPU. If I remember correctly, that portion of the book is fairly short and comparatively easy to understand. Barwise makes his motivations for using KPU clear.

1. I don’t know Hrbacek and Jech’s book at all, and will check that out, thank you. Schimmerling’s little book I forgot about though it is on my shelves, and I’ll remind myself about it, so thanks again.

2. Paul Cohen’s Set Theory and the Continuum Hypothesis is a good reference in the subject, there is also a nice introduction in mathematical logic (first-order logic, Godel’s theorems.). It should appear somewhere in the list.

3. It seems to me none of the books covers the same range of materials as Enderton’s book (which, as you say, costs an arm and a leg). Enderton’s seems perfect for giving beginning graduate students in philosophy the needed basic tools.

Talking to Thomas Forster this summer he suggested a textbook which sounded appropriate, but I did not make a note and then forgot the title.

1. Yes, looking back at Enderton — which I hand’t looked at in decades, not owning a copy myself! — it came back to me what a good introductory book it is. (And good thought: I’ll ask TF what he recommends these days!)

4. A few more titles:

Azriel Levy, Basic set theory, Springer, Berlin, 1979, reprinted with list of corrections: Dover, New York

Kenneth Kunen, The foundations of mathematics, Collegepublications, London, 2009 ( chapter 1, Set theory, pages 9 – 85)

A. Shen and N. K. Vereshchagin, Basic set theory, AMS (student mathematica library, vol. 17), Providence, 2002

Oliver Deiser, Einführung in die Mengenlehre, Springer, dritte Auflage, Berlin, 2009

Frank Drake and D. Singh, Inetrmediate set theory, John Wiley & Sons, New York, 1996

Gregory Moore, Zermelo’s axiom of choice, Springer, Berlin, 1982 (to appear reprinted in Dover, 2013)

5. I’ve always thought Judith Roitman’s Introduction to Modern Set Theory was one of the most interesting intro texts. It used to be absurdly expensive, but there’s now a very inexpensive (and revised) paperback. It’s only £5.77 (!) on Amazon UK.

Another I think is worth considering is George Tourlakis’s Lectures in Logic and Set Theory, Volume 2: Set Theory. (Vol 1 is Logic, but the Set Theory volume can stand alone.) It covers more material than most (including forcing) and seems very readable. It and Enderton are the two books I can recall giving the reader some intuitive insight into cofinality rather than just giving the definition and starting to use it.

(A paperback edition appeared in September 2011 and is available from CUP for £40, but for some reason Amazon UK seems to have stopped stocking it. Strange.)

Re forcing, I remember your post that quoted Timothy Chow calling it an “open exposition problem”. The Tourlakis book looked like one of the better attempts to explain it.

6. More books:

John L. Bell, The Axiom of Choice. (Like the other College Publications books — Kunen’s Set Theory and his Foundations, mentioned above — it’s reasonably priced.)

Set Theory (London Mathematical Society Student Texts), by by Andras Hajnal and Peter Hamburger, translated by Attila Mate. It’s said to be good on combinatorial set theory which takes up roughly the 2nd half of the book.

Set Theory for the Working Mathematician (London Mathematical Society Student Texts) by Krzysztof Ciesielski.

There is, of course, Halmos’s Naive Set Theory (which now has a cheap paperback edition). It’s what I learned set theory from, years ago, but I think it stops just as things are getting interesting and that it’s best suited to people more interested in other things.

(I keep hoping that Drake’s Large Cardinals will appear from Dover, but it hasn’t.)

7. Thinking again of advanced set theory, forcing and independence, there are some more general mathematical logic books that might be worth mentioning:

Peter G. Hinman, Foundations of Mathematical Logic.

Yu. I. Manin’s A Course in Mathematical Logic for Mathematicians (Graduate Texts in Mathematics). This is a revision of his earlier A Course in Mathematical Logic that also has new material on model theory written by Boris Zilber.

Joseph Schoenfield’s Mathematical Logic

And at an introductory level, Wolf’s Tour Through Mathematical Logic (which also introduces large cardinals).

(Model theory texts often have good, relatively advanced material on set theory as well.)

Keith Devlin’s Constructibility is now out of print, expensive and hard to get, though libraries should have it. However, this is still in print: The Axiom of Constructibility: A Guide for the Mathematician (Lecture Notes in Mathematics). Unfortunately, it’s expensive for something so short (104 pages) and set in ‘typewriter’.

Axiom of choice:

Axiom of Choice (Lecture Notes in Mathematics, Vol. 1876), by Horst Herrlich.

We’re fortunate these days to have so many good books available. But that means that people need help on which ones to read, which is one reason your “Teach yourself logic” notes are so interesting and valuable. I’m looking forward to seeing which ones make it into the list.

8. Many thanks for the additional suggestions, a couple of which are new to me. They indeed illustrate the very problem the Guide is trying to address — there is a plethora of books out there and it is difficult to find a good route through.

I’ll return in a later post to say more about all this!

9. Akihiro Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings

I didn’t mention it earlier, because I thought it might be too advanced, but the historical element makes it more accessible, and I think more philosophically interesting, than a purely technical account would be.

One reason I like the idea of two sections is that it might make room for this book to be included.

(It’s available in paperback for around £40, which is not bad for a 536 page book of this sort, and of course libraries should have it.)

10. I have been quite enjoying Potter recently (indeed, I discovered this website as a side effect of searching for electronic versions of some of the bibliographic material therefrom). Having studied “traditional” ZFC from the usual suspects (Kunen and Jech) at Irvine (hello Aldo!), I find Potter’s presentation to be very refreshing, not to mention humorous. This is my second or third time coming back to it since I first read it 4 years ago.

Also I thank you for including Forster. I find this “minority sport” (as Potter aptly put it) of NF and similar to be quite fascinating, and feel it’s a shame more researchers have not had the boldness to work in this field. It seems to have fallen victim to a vicious circle – the theory is usually dismissed because so little has been proven there, but of course perhaps more would be proven if it were not so readily dismissed.

Finally, I thought I had an actual original suggestion, but upon a closer re-read of your post I see you mention it briefly: let me give a hearty “Yes!” vote to including Bell’s Boolean-Valued Models. It’s the only account of forcing I’ve read that truly convinced me, as opposed to all the others which always seem to have some suspiciously nervous hand-waving about countable models of ZFC.

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