The Ehrenfeucht-Fraisse-game of length omega_1
Alan H. Mekler, Saharon Shelah, Jouko Väänánen · arXiv (Cornell University) · 1993
Let (A) and (B) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fra{i}sse-game of length omega_1 of A and B which we denote by G_{omega_1}(A,B). This game is like the ordinary Ehrenfeucht-Fraisse-game of L_{omega omega} except that there are omega_1 moves. It is clear that G_{omega_1}(A,B) is determined if A and B are of cardinality = aleph_3 there are A and B of cardinality kappa such that the game G_{omega_1}(A,B) is non-determined.