Christian Espíndola: Complete theories and their applications

In this occasion we will show in what way metamathematical techniques can help to prove mathematical theorems, giving examples of proofs that are carried over by looking at the properties of complete theories axiomatizing the relevant structures involved. We will first discuss a proof of Ax-Grothendieck theorem, which is based essentially on the completeness of the theory of algebraically closed fields of a given characteristic. Later, we will formulate a proof of the zero-one law for random graphs, based in turn on the completeness of a particular theory of graphs with some extension properties.