The Main Gap in the generalized Borel-reducibility hierarchy

During this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided in two kind:classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored trees. The goal of the talk is to see the classifiable theories case and start the non-classifiable theories case by proving that it is possible to map every element of the generalized Baire, f, into a colored tree, J(f), such that; for every f and g elements of the generalized Baire space, J(f) and J(g) are isomorphic as colored trees if and only if f and g coincide on a club.