This will be part II of the talk on complete L_{omega_1,omega}-sentences with maximal models in (at least) two cardinalities. The talk will be self-contained.

Sample theorems

Theorem: If kappa is homogeneously characterizable and mu is the least such that 2^mu>=kappa, then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities 2^lambda, for all mu=kappa, then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities kappa^omega and kappa.

Theorem (Baldwin-Shelah) If mu is the first measurable cardinal and phi belongs to L_{omega_1,omega}, then no model of phi of size greater or equal to mu is maximal with respect to the L_{omega_1,omega}-elementary substructure relation. Speaker(s): Ioannis Souldatos (University of Detroit Mercy)