Book request: mathematical logic with a semantical emphasis. Suppose I am interested in the semantical aspect of logic; especially the satisfaction $\models$ relation between models and sentences, and the induced semantic consequence relation $\implies,$ defined by asserting that $\Gamma \implies \varphi$ iff whenever $M \models \Gamma$ we have $M \models \varphi$. Suppose, however, that I am not currently interested in any of the following (admittedly very important) issues.


*

*Founding mathematics.

*Formal systems, and the limitations of formal systems.

*Recursive axiomatizability.

*Computability theory.


Given my particular mathematical interests, can anyone recommend a good, fairly elementary mathematical logic book?
 A: You'll find an annotated guide to model theory (in fact two guides, an initial pass through elementary books, and then a more advanced guide) in the general reading Guide to logic text books you can download here: 
http://www.logicmatters.net/resources/pdfs/TeachYourselfLogic9-4.pdf
That should give you enough information to be able to send you to the right books in the library given your particular interests.
Given the way the question is phrased, you might possibly also be interested in this side note:
http://www.logicmatters.net/resources/pdfs/Galois.pdf
A: I'm not sure exactly what you're looking for or exactly what your background is, but the first time I saw model theory was with Model Theory: an Introduction by David Marker. I think it might be something you are interested in. 
A: The book "Éléments de Logique Mathématique" by Kreisel and Krivine (which I believe has an English translation, probably with the obvious title "Elements of Mathematical Logic") takes a fiercely semantical approach to the basic parts of mathematical logic.  The material you don't want, about the axiomatic method and foundations of mathematics, is relegated to a couple of appendices.
