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Next semester I will be teaching model theory to master students. The course is designed to be "soft", with no ambition of getting to the very hardcore stuff. Currently, this is the syllabus.

Syllabus. First-order languages. Theories and their models. Compactness and Completeness. Loweinheim-Skolem theorems. Back and forth and countable categoricity. (Galois) Types. Quantifier elimination (?). Saturation. Ultraproducts.

Some big classics contain these topics but shuffle them in ways that I find cumbersome. For example, Poizat presents the back-and-forth almost at the beginning of the book. Other books do quantifier elimination before any form of completeness theorem for FOL. Now, it's not like I think any of these choices are wrong, it's just not the choice I would personally find natural. So I would rather a book that, at least (and especially) for the first chapters, follows the structure of the Syllabus above.

Q. Can you recommend a reference?

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    $\begingroup$ I highly recommend checking out Peter Smith's Logic Study Guide (logicmatters.net/tyl), which contains a nice overview of many mathematical logic textbooks including model theory textbooks. $\endgroup$
    – Alvaro Pintado
    Commented Jun 7 at 12:18
  • $\begingroup$ @AlexKruckman, your notes look great! Would you update your comment to an answer? $\endgroup$
    – Ivan Di Liberti
    Commented Jun 7 at 14:59
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    $\begingroup$ Is someone besides me still waiting for Rami Grossberg's three volume model theory book? (+20 years) $\endgroup$
    – Matemáticos Chibchas
    Commented Jun 7 at 16:57
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    $\begingroup$ I noticed Marker isn't among the answers so far. Any particular reason not to recommend it? $\endgroup$
    – Greg Nisbet
    Commented Jun 9 at 17:32
  • $\begingroup$ @GregNisbet Probably because Marker does not cover the completeness theorem, which seems to be important to the OP. I think Marker is a great book to use for a first course in model theory, and I've edited my answer to clarify that. $\endgroup$
    – Alex Kruckman
    Commented Jun 14 at 15:37

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One very nice recent book: Moerdijk and van Oosten, Sets, models and proofs, 2018 — homepage on Springer.

Unlike most books mentioned in other answers so far, it’s a general introduction to logic, not specifically to model theory. As such, it doesn’t go deeper into model theory specifically; but it covers all the topics you mention except for saturation. It takes roughly the same order you name; ultraproducts come earlier (as an alternative proof of compactness) but are not relied on by anything afterwards. Generally, it has good taste in its presentation, never using power tools unnecessarily (even if they’re already available) where a more elementary argument naturally suffices. It also has good and plentiful exercises, and is breezy and readable throughout.

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I have some notes on model theory from a graduate class I taught at Indiana University. I covered all the topics you listed except ultraproducts. Many of the students in the class were not specializing in logic, so the notes might be at the right level for your masters course. They are available here: https://akruckman.faculty.wesleyan.edu/files/2019/07/Lecture-Notes.pdf

The main distinguishing feature of my notes is that I allow many-sorted logic and empty sorts from the beginning. I also emphasize the role of Stone duality throughout. As Joel says in a comment, the completeness theorem is not really relevant to model theory, so most model theory textbooks avoid defining formal proof and instead prove the compactness theorem directly. But I did prove completeness in these notes, mostly as an excuse to work out a proof system and proof of completeness that works for many-sorted logic with empty sorts and empty structures allowed.

Edit: Since this answer is getting so many upvotes, let me clarify that I am not in any way suggesting that my notes are better than the books suggested in the other answers. I think that the classics by Hodges and Marker are still the best places to learn model theory, complemented by the book by Tent & Ziegler (which is my favorite, but which I think is less suitable for a first course). The main reason I suggested my notes is that the OP specifically asked for a book that covers the completeness theorem, and this is absent from Marker, Hodges, and Tent-Ziegler.

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Jonathan Kirby's book An Invitation to Model Theory (Cambridge University Press, 2019) might fit your needs. I haven't taught from it, but it's designed to be gentler than most texts on the subject. You might also find it more in tune with your own categorical sensibilities than some other introductions to model theory.

It's quite short, but it looks like it covers all the topics you list except for ultraproducts.

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    $\begingroup$ I was checking its table of contents yesterday and I was a bit surprised to not see a section titled "Completeness theorem". Currently it is still the best I have seen indeed. $\endgroup$
    – Ivan Di Liberti
    Commented Jun 7 at 12:19
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    $\begingroup$ @IvanDiLiberti Do you consider the completeness theorem part of model theory? It seems instead to be proof theory, with little connection to the rest of model theory. Further, the standard proofs of completeness are often better taken as proofs of the compactness theorem, which is relevant for model theory. I believe this is why many contemporary model theory texts do not cover completeness or instead proofs systems at all. $\endgroup$
    – Joel David Hamkins
    Commented Jun 7 at 12:45
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    $\begingroup$ @JoelDavidHamkins it is not my intention to embark myself in any form of true Scottish men debate. We all know that there are no true Scottish men. I find your point peculiar, but I am aware of projecting my personal bias. $\endgroup$
    – Ivan Di Liberti
    Commented Jun 7 at 13:08
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    $\begingroup$ I don't really follow your reaction. I was just trying to explain why it is quite common these days to drop completeness in a math logic class, especially one aiming at model theory, and so one should not be surprised by this in the way that you had expressed. $\endgroup$
    – Joel David Hamkins
    Commented Jun 7 at 13:17
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I quite like Wilfrid Hodges's A Shorter Model Theory (Cambridge University Press, 1997). It covers all the topics you mention, while also tackling a few more advanced ones in the final chapter. The book teaches via a good stock of examples, mostly (as one would expect) from algebra. It doesn't go too fast but should contain enough to stretch stronger students. I first read it as a Master's student myself, in a reading group with my supervisor and a few other graduate students at various levels. This was in a philosophy department and I think by the end only three of us were still following, but I got a lot out of it. The exercises are good, because they're generally hard enough to really learn from, at least for a student who's seeing the material for the first time.

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After some study of Peter Smith's Logic Study Guide, as suggested by Alvaro Pintado in the comments, I have settled on the following two books.

  • Manzano, Model Theory.
  • Delzell and Prestel, Mathematical Logic and Model Theory.

I am still very much open to other answers and I will accept this one only in a couple of weeks or so.

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Rautenberg's "A Concise Introduction to Mathematical Logic" is my recommended introduction to a proper study of logic, including all the basics as well as a little bit of proof theory and model theory. This book alone should be enough if you are not going to do any really difficult stuff, as it covers semantic completeness, compactness, Lowenheim-Skolem, quantifier elimination, Vaught's test, model completeness, Ehrenfeucht games, and states but does not prove Morley's categoricity theorem.

For model theory specifically, there are many good free online references such as Anand Pillay's (especially for Morley's theorem), George McNulty's and William Weiss' & Cherie D'Mello's.

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