Compactness and Complete Theories

The Compactness theorem and some of its consequences are studied. The Upward Löwenheim–Skolem is proved and used to deduce Vaught’s result on the completeness of categorical theories. We show that the full theory of the field of complex numbers is axiomatized as the theory of algebraically closed fields of characteristic zero and give several consequences. Countable categoricity allows us to study dense linear orders and random graphs.