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?

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.
