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I have been encountering many results on logic-related properties of algebraic structures such as elementary equivalence, axiomatizability, definability, etc. The problem is that when I see the proof or a sketch of the proof, I understand every third word or less. What should I read to get proper understaning of the methods used in such proofs? I would most apperciate a book focused mainly on algebra and first order logic.

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Any book with "model theory" and "introduction" in the title should work. – Chris Eagle Jan 19 '12 at 18:53
+1 Chris however, personally I found delving straight into model theory can be a bit difficult and a text such as Geoffrey Hunter's Metalogic can ease up the transition. – Sniper Clown Jan 19 '12 at 19:07
up vote 3 down vote accepted

A useful (and free!) book is Fundamentals of Model Theory by Weiss and D'Mello.

I second the recommendation for Hodges' Model Theory. It is long-run more useful than his Shorter Model Theory.

Marker's Model Theory is also very good.

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Thank you very much! – user23211 Jan 19 '12 at 19:23
I second the suggestion of Marker's Model Theory: An Introduction. It is especially focused on algebraic structures. – Quinn Culver Jan 22 '12 at 2:22

An excellent choice is Wilfred Hodges textbook Model Theory, which goes into much further detail than most alternatives, and has many illuminating notes that will help you to "think like a model theorist" (and some good jokes to boot!) Also extremely useful are the many survey articles in the Handbook of Mathematical Logic (edited by J. Barwise)

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Thank you very much! – user23211 Jan 19 '12 at 19:23

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