## 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:

- Back to the beginning
- Getting to grips with first-order logic
- Modal logic
- 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.

Suggestions please!

*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”.