Ravens, so we read, can only count up to seven. They can't tell the difference between two numbers greater than or equal to eight. First-order logic is much the same as ravens, except that the cutoff point is rather higher: it's ω instead of 8.

This chapter is wholly devoted to the model theory of first-order languages. First-order model theory has always been the heart of model theory. The main reason for this is that first-order logic, for all its expressive power, is too weak to distinguish between one large number and another. The result is that there are a number of constructions which give models of a first-order theory, or turn a given model into a new one. In this chapter we study two such constructions. The first is a combination of the compactness theorem with (Robinson) diagrams. The second is amalgamation; it can be seen as an application of the first.

Granted, some fragments of first-order logic are even weaker than the full logic, so that they allow even more constructions.