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What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am interested in arithmetic geometry, is there a reference with a view towards such a topic?

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If you don't have much background in logic start with a general logic book that has a chapter on model theory, e.g. this or this. –  Kaveh Jun 23 '12 at 4:30
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For the curious and lazy, @Kaveh's two links are to Shoenfield, Mathematical Logic, and Mendelson, Introduction to Mathematical Logic. –  Henning Makholm Jun 23 '12 at 17:44

9 Answers 9

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I really like Introduction to Model Theory by David Marker. It starts from scratch and has a lot of algebraic examples.

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Really? I saw it when I search on amazon but most of the reviews said it was rather terse and that you needed much background. That was not your experience then? –  Eugene Jun 23 '12 at 3:53
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@Eugene I think that someone who has taken at least a year or two of undergraduate mathematics should have the background to understand most of the examples. And of course, you can always skip the examples, if what you are interested in learning the logic. Of the model theory books I know, I think Marker's is the easiest and most modern. Chang and Keisler and Hodge's book are full of results that may be useful for a researcher, but the books are harder to read and moves so much more slowly. –  William Jun 23 '12 at 4:42
    
@Eugene Why don't you take a look at the first chapter? –  azarel Jun 23 '12 at 12:59
    
FWIW, I have found Marker's book (which I read as a graduate student, with zero prior exposure to the subject) to be a significantly easier and quicker read than most other model theory texts I have read, or tried to read, since then. –  Pete L. Clark Aug 4 '12 at 7:26

For a free alternative, Peter L. Clark has posted his notes Introduction to Model Theory on his website. He says no prior knowledge of logic is assumed and the applications are primarily in the areas of Algebra, Algebraic Geometry and Number Theory.

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Besides Marker, for basic model theory I also recommend Barwise's compilation "Handbook of mathematical logic", Chang, Keisler & Troelstra's "Model theory" and Wilfrid Hodges' "Model theory" (he's also written "A shorter model theory", but I haven't seen it.).

However, I've only ever used them as reference, not as an actual textbook, so I can't guarantee their quality as such.

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You could give Bruno Poizat's A Course in Model Theory a try.

If you are feeling particularly ambitious, perhaps Model Theory and Algebraic Geometry (E. Bouscaren, ed.), which intends to gives an introduction to certain concepts in the interplay of model theory and algebraic geometry, with a view to an exposition of Hrushovski's proof of the geometric Mordell-Lang Conjecture. If nothing else, this work should give an idea of what concepts of model theory have found application in algebraic geometry (at least in the aforementioned proof), which should give you an idea of perhaps what topics to look for in a model theory text.

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Also, as I am interested in arithmetic geometry, is there a reference with a view towards such a topic?

One paper I was able to find was Model Theory and Diophantine Geometry by Anand Pillay

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There are a bunch of suggestions on Peter Smith's blog.

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I second Peter Smith's enthusiasm for Wilfrid Hodges. He's my favorite. If I am looking for exposition on a topic, I try to find something by Hodges first. He's knowledgeable and hysterical and will tell you lots of little useful things that most books won't mention. If you try his model theory book but don't feel ready for it, he wrote a little gem of a logic book, Logic. –  Rachel Jun 23 '12 at 23:49

These notes by Stefan Geschke have been extremely helpful to me. I currently taking the course these notes are based on and often I find the exposition (and the proofs) here much more illuminating than Marker's text. The goal of these notes is to prove Morley's Categoricity Theorem. If one is already using a tradional text, such as Marker, these notes are at least a very good companion to Marker.

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I found these notes by Stephen Simpson very useful.

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Fundamentals of Model Theory by Weiss and D'Mello is available for free download.

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