ABSTRACT
As a first application of the finiteness theorem we derive a fundamental result about arbitrarily large models. Then we give some simple applications to classical structures. In order to apply the finiteness theorem to these, we have to first axiomatize them in an appropriate (first-order) language. Finally, we explain why the finiteness theorem is also called compactness theorem. This leads to certain topological spaces, the so-called Stone spaces, which we will return to later in Part IV.1
