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I am looking for a good book about model theory. As this is obviously too vague, let me explain what I am looking for and why.

First I am interested about the basics and foundations of model theory. Right now I am not interested in their applications (like proving things in mathematics or even independence results like Cohen's -- but of course it is not a problem if a book deals with some of these applications if it does not only that).

Second, until a few days ago I believed I knew well enough what was a model. But since two days I am not so sure. I have a problem with the notion of model inside ZF (or ZFC, or any formalized set theory) of a theory, and in particular, with the meaning of the satisfaction relation in this case. I would like a book which treats that aspect.

These problems arose with my trying to understand the answers to my question A meta-mathematical question related to Hilbert tenth problem

I am currently having endless discussions in comments about that notion what I am said doesn't make sense to me (and the converse is clearly also true). A good book will certainly help me and save time for my respectable interlocutors. Thanks...

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  • $\begingroup$ Reading the MO question you linked, I think you're just interested in becoming aware of some simple conventions (assumed as understood by logicians when dealing with "formal theories") that lie not so deep but are just a matter of the "bureaucracy" of mathematical rigour. Conventions that often confuse us non-logicians. $\endgroup$
    – Qfwfq
    Commented Sep 28, 2011 at 16:18
  • $\begingroup$ @unkowngoogle. That's true for the most part : the only way for me to make sure that my misunderstandings with the logicians/model-theorists in the MO question I linked is just a matter of bureaucracy is to really understand enough of that bureaucracy. But I am also genuinely interested in learning more about models. $\endgroup$
    – Joël
    Commented Sep 28, 2011 at 16:29

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Basic model theory texts are Marker's Model Theory; An Introduction and A Shorter model theory by Hodges. Maybe the one on Mathematical Logic by Cori and Lascar too. I'm not sure you need a book which specifically treats this aspect but a general understanding of what a theory, and a model of a theory (e.g. ZF or ZFC) is should do (the first chapter of Marker's book covers this). Once you know the basics then just think of a language with a binary relation symbol (for set membership) and formulas (in the language) for the axioms of ZF. Then you can think of a model of this etc...

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  • $\begingroup$ When I took Model Theory the required textbook was [Wilfred Hodge's A Shorter Model Theory.][1] I found it to be a very clear and concise introduction to the key theorems in the subject for someone who has had some basic Abstract Algebra at the graduate level. [1]: amazon.com/Shorter-Model-Theory-Wilfrid-Hodges/dp/0521587131/… $\endgroup$
    – J.L. Nelson
    Commented Sep 29, 2011 at 0:38
  • $\begingroup$ Hodges is a bad choice in this case, since it has much less about models of PA (look at the question that the poster links to). Model theory reference have been done on MO, and I assume people can search. So, if this is to be a meaningful and new question, I think people should answer in the context of the particular poster's needs. $\endgroup$
    – James Freitag
    Commented Sep 29, 2011 at 13:32
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For understanding the referenced MO question, are you sure that you want a book on Model theory? I think that a request for a book introducing first order logic is what this post is really asking for. I won't make suggestions for learning basic first order logic, but perhaps others could.

In case I am wrong, and the poster is really interested in learning model theory. I don't think I see the point of answers which simply list model theory texts and have little to do with the specific reference request. Any model theory book will give the poster the basics. From the question, a book emphasizing models of PA would be a better suggestion. Here are some details about the specific books:

Marker's book covers PA in several chapters based on the model theoretic techniques used and contains a fair number of exercises on models of PA.

Poizat's book contains much less (almost nothing technical), so it would not seem to be a good suggestion.

From what I recall Marcja and Toffalori contains even less (although I haven't looked at this much, so I could be wrong in this recollection).

Hodges contains less than Marker and more than Poizat, but is also not a good choice for this poster's goals.

I have not read Cori and Lascar, so I can not comment definitely, but I will mention that the table of contents at least mentions the Peano axioms. Also, this suggestion would seem to be in line with what I wrote in the first paragraph.

The above books were included (in part) because other people mentioned them in their answers. The following should be mentioned, but this is not a comprehensive list. Hodges Model theory (not the shorter one...) has more PA than his shorter book, but still not as much as Marker's book. Kaye's Models of Peano Arithmetic is hard to find (at least it was a few years ago), but the parts of it that I have read seemed well-written.

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  • $\begingroup$ No, I am seriously interested in learning model theory for its own sake, beyond the basics (Godel's completeness theorem, the compacity theorem, etc.) I learnt when I was in the French equivalent of a "graduate school". The foundational problems I had are solved now. I still think that the text books on model theory quoted here (or the ones in planet math and other sites) which all define the notions of models, theory, of language using sets, should say explicitly if they mean "sets" as intuitive objects or as objects of a formal theory like ZFC. They are all (excepted Cohen) mute about this $\endgroup$
    – Joël
    Commented Sep 29, 2011 at 16:21
  • $\begingroup$ Thanks to all for your suggestions. I have looked at most of them (the Cori-Laskar was the one I used back in grad school). James, I agree with you, the best for me seems to me Marker's book. $\endgroup$
    – Joël
    Commented Sep 29, 2011 at 16:23
  • $\begingroup$ Alright, I just want to caution you that Marker's book is great for those wishing to study modern applied model theory, but I have gone through most of the book while rarely thinking about the sorts of foundational worries you have brought up here and in your other posts. Perhaps that is just a function of taste. I read the book thinking about mathematical applications. Maybe it will be different if you read it from a different perspective. $\endgroup$
    – James Freitag
    Commented Sep 30, 2011 at 1:03
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Based on our discussions in your other MO question, I believe that what you want to see is not a book about model theory, but a tutorial about how to formalize ordinary mathematics in ZFC, with model theory being a specific case of interest. One resource you might find useful is an article by Leslie Lamport in which he takes you through an example slowly (the Riemann integral I think). You should be able to find it by Googling "formalizing mathematics lamport". Once you get the general idea, you should be able to apply it to other cases.

The only confusing thing about model theory specifically, I think, is that in model theory one works with formal languages, which don't show up in "classical" (19th century or earlier) mathematics. But if you encode symbols as sets, and strings as sequences of symbols, then there should be no problem.

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  • $\begingroup$ Timothy, actually I have no serious problem about how to formalize mathematics, the theory of the Lebsegue integral for example, in ZFC. The problem I had about model theory in ZFC (and that you helped me to solve) is that at first, I was thinking that only the model itself was considered as a formal object of ZFC, while the language and theory of which one considers a model was still "in the spirit of traditional mathematics, outside any formal language", as Paul Cohen puts it in his book "set theory and the continuum hyp." This leads me to problem that you don't have if you work either... $\endgroup$
    – Joël
    Commented Sep 29, 2011 at 16:13
  • $\begingroup$ in a pure non-formal way, for both the theory and its model (as Cohen does in the beginning of his book), or entirely in ZFC. $\endgroup$
    – Joël
    Commented Sep 29, 2011 at 16:14
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Brunot Poizat's "A course in model theory" or Marcja&Toffalori's "A guide to classical and modern model theory" come to mind.

If you need very very basic things about models, Cori&Lascar "Logique mathématique" has an english translation.

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  • $\begingroup$ To be clear, I voted down this answer, because I feel like you are simply listing logic or model theory texts rather than answering the question based on the poster's request. I am not judging the books listed, though I think the two model theory texts would be particularly inappropriate given that the poster specifically is interested in models of PA (from the referenced MO question). $\endgroup$
    – James Freitag
    Commented Sep 29, 2011 at 4:45
  • $\begingroup$ @James Freitag: Well, I'm first listing two reference texts which deal with model theory, but I'm none too happy about them precisely because they're too focused on model theory itself. I'm convinced the last reference, C&L, which only barely touches models at the very end, will answer the question nicely, precisely because it doesn't cover models for themselves, but introduces and defines them at an elementary level, with nice definitions. After all, the poster wrote that (s)he wasn't too confortable with the notion of model inside a theory... @Joël: It's Lascar, not Laskar, beware! $\endgroup$
    – Julien Puydt
    Commented Sep 29, 2011 at 12:27
  • $\begingroup$ Right, but this is a poster who asked a question which was essentially about understanding models of PA, and you listed two books which barely cover this while there are perfectly suitable options which cover the topic while also covering basic model theory (and first order logic, which is what he really seems to be after). (I would vote up your answer if you explained what about the choices made you pick the particular books. But you would have to convince me why this poster should choose the Poizat book, for example.) $\endgroup$
    – James Freitag
    Commented Sep 29, 2011 at 13:30
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Rather than to a book, I point you to real formalizations in a set theory: I deem this appropriate given the question itself. Otherwise, downvote me please :)

I happen to have formalized in Mizar set theory (which is Tarski-Grothendieck, i.e. ZFC on steroids) the stuff you seem pointing to: language, wffs, interpretation, satisfaction relation, evaluation, sequent derivability, provability, etc...

In FOMODEL1: http://mizar.auburn.edu/version/current/html/fomodel1.html

you get most syntax (up to definition of atomic formula, or 0wff).

In FOMODEL2: http://mizar.auburn.edu/version/current/html/fomodel2.html

you will find the definition of satisfaction.

That is a series of five subsequent articles starting from scratch and getting to completeness theorem (and Lowenheim-Skolem, the latter only on my homepage, not submitted to Mizar people yet). The links point to hypertextual, proof-pruned versions. For full formalizations, look for the same files with the extension .miz in that same server.

Mizar formalizations are arguably among the most readable for the average mathematician (that's the factor that got myself started with it), that's why I thought you could find this stuff of some interest.

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I might be completely mistaken, but I dare say you should take a look at set theory first, maye that could clarify your questions. I'd recommend the first four chapters of Kenneth Kunen's book, at least they helped my understanding, but maybe that's not the thing you had in mind. In particular:

Chapter 1, §14: "Formalizing the metatheory"

Chapter 4, §9: "Model theory in the metatheory"

Chapter 4, §10: "Model theory in the formal theory"

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