# Simple book on model theory

I was expressed by how Mendelson describes models in his Introduction to mathematical logic. Now I am looking for a nice model theory guide. The book (video source, etc.) must:

1. Include the concrete methods with their proofs and must answer the following questions:

1.1. how to know if a theory has a model
1.2. how to build a model if a theory is consistent
1.3. how to know if a class of structures forms the models of some theory
1.4. how to build a theory for an elementary class
1.5. given a structure, what information can be obtained by logic

2. Be concentrated on finite models and theories (that's why Keisler doesn't fit)

3. Not contain too much algebra (that's why Marker doesn't fit)

4. Not contain complexity theory at all (that's why Ebbinghaus doesn't fit)

So I am interested in a simple guide containing necessary proofs. If there is no such a book, is it real to discover the methods above by myself?

• Concerning a few of your criteria: 1.1. It is algorithmically undecidable whether a theory (or even a single sentence) has a model. 1.2. A theory that has models need not have computable models. 2 It is algorithmically undecidable whether a theory (or even a single sentence) has a finite model. (My opinion about 4: Finite model theory and complexity theory are nearly the same thing.) In other words, model theory is neither as simple nor as isolated from other fields as you want it to be. – Andreas Blass Aug 10 '19 at 10:41
• Thanks for your answer! Actually, I realize the difficulties you've mentioned. I think I want to learn about special (computable) cases and methods of the model-building. – Elmar Guseinov Aug 10 '19 at 12:04
• In a finite language any collection of finite structures is the class of finite models of some theory. – James Hanson Aug 10 '19 at 12:27
• Wilfrid Hodge's homepage is well worth a look, too. – James Smith Aug 11 '19 at 10:02
• It seems that Trakhtenbrot's Theorem would prohibit such a book from ever being written. That said, much of the theoretical literature in database theory seems to have similar objectives that you have. Maybe? – François G. Dorais Aug 15 '19 at 4:40

The best book for you is probably A Shorter Model Theory by Hodges.

Some comments on your question, though: First, you should be aware that the model theory of finite structures and the model theory of infinite structures have extremely different characters - so much so that finite model theory is essentially a separate subfield of logic, which is much closer to computer science and complexity theory. This can be (partially) explained by the fact that first-order logic is powerful enough to completely describe finite structures, so interesting questions in the first-order model theory of finite structures have to impose some constraints: working with fragments of first-order logic and taking complexity into account.

If you're really interested in finite model theory, you can take a look at this question, which has some references in the comments and answers. To my knowledge, the book by Ebbinghaus and Flum is the textbook on the subject which contains the most material not directly related to complexity theory (though there are probably books that I'm not aware of).

On the other hand, "ordinary" model theory is primarily concerned with infinite models, and as a result it's hard to avoid some set theory creeping in. If you're really turned off by ordinals and cardinals, I would recommend: (1) learn something about them, set theory is a beautiful subject! (2) in the mean time, concentrate on the model theory of countably infinite structures. This is a domain in which you get to see many of the concepts and techniques of model theory at work without any transfinite inductions in sight (except in more advanced topics: the Scott rank and Morley rank can be useful for studying countable structures, and they are both ordinal-valued).

It's also the case that many of the interesting examples in model theory come from algebra. So it's hard to achieve your requirements 2, 3, and 4. But this is why I suggested Hodges: In my experience students without a strong background in algebra and set theory find Hodges's book to be easier to read than Marker's.

I offer you to try these books too:

-A Guide to Classical and Modern Model Theory, Written by Annaliza Marcja and Carlo Toffalori. I could find many tangible examples in this book when I was trying to understand Marker's.

-A course in Model Theory, Katrin Tent and Martin Ziegler. Although one can find most of the chapters in Marker's book, but some chapters are written more simple seemingly.

• I especially like the book by Tent and Ziegler (which in its second half goes far beyond the material in Marker's book). But I don't think it's suitable as a first introduction to model theory - it's very terse, and there are few examples. – Alex Kruckman Aug 10 '19 at 15:16
• Oh, I can't find it but it seems to be interesting. – Elmar Guseinov Aug 10 '19 at 17:09
• "A Course in Model Theory (Lecture Notes in Logic Book 40)" by Tent and Ziegler, ISBN 978-0521763240, can be found for sale online (or in bookstores, I guess). Also for free as pdf at certain websites, but just downloading it without paying could be illegal, so I don't think posting their address would be ok. – Goldstern Oct 19 '19 at 13:13