Semantics of First-Order Languages

In the previous chapter, we presented the syntactical notions pertaining to first-order theories. However, in general, mathematical theories are not developed syntactically. There is, of course, one serious exception to this: essentially, due to its foundational nature, axiomatic set theory is developed syntactically. Since set theory is needed for proving independence results, the syntactical approach is quite important for mathematics. In this chapter we give the semantics of first-order languages to connect the syntactical description of a theory with the setting in which a mathematical theory is generally developed.