# Teach yourself logic, #3: Beginning model theory

I have been working away on the second edition of my Gödel book. The current task: giving a more lucid proof showing Robinson arithmetic can represent all primitive recursive functions. In the first edition I cheated by taking a clever trick from Burgess, Boolos and Jeffrey. I do now regret that. But I can certainly sympathise with my earlier self for taking the easy way out!

By way of diversion, then, and as an exercise in constructive procrastination, here is the draft third instalment of my slowly developing ‘teach yourself logic’ guide. So far we’ve covered (1) standard first-order logic, at an introductory level, and (2) some basic modal logic. Today’s new list (3) looks at the path forward from what’s covered in a standard first-order logic course on towards full-blown model theory. [The ordering of the instalments here is going to be henceforth a bit arbitrary; but I hope a tolerably sensible structure will emerge in the final composite Guide!]

In fact, in reworking the first two instalments of the Guide — you can now download an expanded version of them here — I have rethought the division between what is to go in instalment (1) and this new one. So take the treatment of first-order logic in (1) now to get just as far the completeness proof but really no further (so, that’s pretty much the content of e.g. Chiswell and Hodges’s terrific Mathematical Logic).

So where next, if you want to move on from those first intimations of classical model theory in the completeness theorem to something of a grasp of the modern theory? There is a very short old book, the very first volume in the Oxford Logic Guides series, Jane Bridge Beginning Model Theory: The Completeness Theorem and Some Consequences (Clarendon Press, 1977) which takes on the story just a few steps pretty lucidly. But very sadly, the book was printed in that short period when publishers thought it a bright idea to save money by photographically printing  work produced on electric typewriters. So, used as we now are to mathematical texts beautifully LaTeXed, the look of the book is decidedly off-putting. So let’s set that aside (as the first recommendation covers much of the same ground anyway).

Here, then, are two natural and rather complementary places to start:

1. Dirk van Dalen Logic and Structure (Springer 4th edition 2004). In instalment (1) I warmly recommended reading this modern classic text up to and including Section 3.1, for coverage of basic first-order logic. Now read the whole of Chapter 3, for a bit of revision and then for the Löwenheim-Skolem theorems and some basic model theory.
2. Wilfrid Hodges’s `Elementary Predicate Logic’, in Handbook of Philosophical Logic, Vol. 1, ed. by D. Gabbay and F. Guenthner, (Reidel 2nd edition 2001). This is an expanded version of the essay in the first edition of the Handbook, written with Hodges’s usual enviable lucidity. Over a hundred pages long, this serves both as an insightful and fresh overview course on basic first-order logic (more revision!), and as an illuminating introduction to some ideas from model theory.

For a more expansive treatment (though not really increasing the level of difficulty, nor indeed covering everything touched on in Hodges’s essay) here is a still reasonably elementary textbook:

1. Maria Manzano, Model Theory (OUP, 1999). I seem to recall, from a reading group where we looked at this book, that the translation can leave something to be desired. However, the coverage as far as it goes is good, and the treatment accessible. I like the way it starts off by talking about relationships among structures in general before talking about structures-as-models-of-theories.

This might already be about as far as most philosophers will want to go. But if you do press on, the choice at the next level up is surely self-selecting:

1. Wilfrid Hodges A Shorter Model Theory (CUP, 1997). Deservedly a modern classic — under half the length of the encyclopedic original, but still full of good things, going a good way beyond Manzano. It gets tough as the book progresses, but the earlier chapters should be manageable.
2. Rather different in focus is another older book, which is particularly elegant (though perhaps you will need more mathematical background to really appreciate it) is J. L, Bell and A. B. Slomson’s Models and Ultraproducts (North-Holland 1969; Dover reprint 2006). As the title suggests, this focuses particularly on the ultra-product construction.

Finally, though probably this is looking over the horizon for most readers of this list, at a further notch up in difficulty and mathematical sophistication, there is another book which has also quickly become something of a standard text:

1. David Marker, Model Theory: An Introduction (Springer 2002). Rightly very highly regarded. (But it isn’t published in the series ‘Graduate Texts in Mathematics’ for nothing!)

So that is my main list. What have I missed out? Well, you could still get a lot out of C. Chang and H. J. Keisler’s classic Model Theory (North Holland, 2nd edition 1977). This is leisurely, very lucid and nicely constructed with different chapters on different methods of model-building. You could well look at quite a bit of this before or alongside reading Hodges’s book. There’s a short little book by Kees Doets Basic Model Theory (CSLI 1996), which concentrates on Ehrenfeucht games which could appeal to enthusiasts. And then, of course, many Big Books on Mathematical Logic have chapters on model theory: a good treatment of some central results seems to be that in Shawn Hedman, A First Course in Logic (OUP 2004), Chs 4–6 which could be perhaps read after (or instead of) Manzano.

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### 10 Responses to Teach yourself logic, #3: Beginning model theory

1. David Auerbach says:

Off-topic (though you started it): Just wondering why you don’t want to use the “trick” in B&J. Is it that it hides something that should be overt? Or is it that it avoids a lesson in technique that would be useful? Or is it just that it has the aura of trick?

• Peter Smith says:

At least in the version given in my book, it does have the aura of a trick, methinks!

2. Rowsity Moid says:

Chang and Keisler should 3rd edition should be appearing from Dover soon, making it very affordable as 600+ page logic books go. Amazon US already has it in stock for \$22.83. (The final North Holland edition was also the 3rd. Do you prefer the 2nd, or will the 3rd do?)

At that price, it is very appealing. By comparison, “longer Hodges” costs £75 in paperback and a ridiculous £172.90 as a hardcover.

Bell and Slomson’s Models and Ultraproducts is also available from Dover. (When I was a maths student, the model theory course alternated between this book and Chang and Keisler, depending on which was most reasonably available that year.)

Bruno Poizat’s A Course in Model Theory is interestingly different and though normally quite expensive is currently part of the Springer “Yellow Sale”.

Introduction to Model Theory by Philipp Rothmaler looks like it might be pretty good.

There’s a book, A Guide to Classical and Modern Model Theory, by Annalisa Marcja and Carlo Toffalori, that covers a lot of material, but I haven’t looked at it in any detail.

Sacks’s Saturated Model Theory (another classic text) is back in print.

Admittedly, some of these aren’t quite beginning model theory, but then neither is Marker, and they include the beginning parts.

• Peter Smith says:

Many many thanks for this!

1. Chang and Keisler. Ah, I just reached for the copy on my shelf and didn’t think to check that whether there was a later edition. Good news about the Dover reprint. And I was wavering whether to put it in the main list, and on reflection I think that’s where it belongs.

2. Bell and Slomson. Yes. Great book. I have to plead a senior moment here, as it was on my desk as I was typing the blog entry, and I meant it to appear on the list, now revised to include it!

3. Poizat. Well I guess if I’m going to mention Marker’s hard-core book, this should get a mention too. (I have to confess I’m making up the criteria for inclusion as I go along!)

4. Rothmaler I forgot about, and my eye skipped past it as I was looking along my selves for inspiration and reminders. I’ve not studied it carefully, though I recall it as being clearly written. I’ll look back at that.

5. Marcja and Toffalori is a new one to me. I will take a look next time I’m in the library.

6. Sacks I thought of as being a step or three further on again.

3. AZ says:

Very nice post!

I just would like to add two more books on model theory. Unfortunately, I haven’t read any of them.

- Alexander Prestel & Charles N. Delzell «Mathematical logic and model theory. A brief introduction», Springer.
- G. Krisel & J.L. Krivine «Elements of mathematical logic. Model theory», North-Holland.

• Peter Smith says:

Thanks for this. The old and the new! I remember Kreisel and Krivine’s 1967 book as being interesting in parts (if you already know an amount) but certainly not an easy read.

Prestel and Delzell’s new 2011 book looks interesting “a streamlined yet easy-to-read introduction to mathematical logic and basic model theory” says the blurb and it is less than 200 pages long too. I will check it out! Does anyone already know this book?

4. Konstantinos Pouliasis says:

Shoenfield’s book is a classic. I think it deserves to be mentioned on your lists.
Best,
Konstantinos

• Peter Smith says:

I agree: Schoenfield is a real classic. I’ll have to think how and where to mention it, but you are quite right. It should most certainly feature somewhere in the developing lists.

5. Rowsity Moid says:

Thoughts of “typewriter” books and Dover intersected here: Hodges’s Building Models by Games. I wouldn’t call it introductory, but it’s a different sort of take on the subject and it may belong in the lists somewhere. (In effect, it’s about forcing.)

But that reminded me of this recent book: Models and Games (Cambridge Studies in Advanced Mathematics) by Jouko Väänänen. Here’s one of the review quotes on Amazon:

“This is a very valuable book, written by a highly competent logician and mathematician who has himself contributed to the field. Parts of it (roughly, chapters 1 to 7) can be used in an introductory course in model theory with a game-theoretical flavor. The last two or three chapters, however, will require a bit more mathematical bravery, but the effort pays off.”

Walter Carnielli, Computing Reviews

I bought a copy a while back, haven’t yet had a chance to read it properly, but it looks interesting and not too immediately difficult.

6. Nick Smith says:

Perhaps worth a mention here: the first part of Barwise ed. Handbook of Mathematical Logic — especially the chapter by Keisler.

Cheers,

Nick