THE EHRENFEUCHT-FRAISSE-GAME OF LENGTH OMEGA(1)

Let U and B be two first order structures of the same vocabulary. We s hall consider the Ehrenfeucht-Fraisse-game of length omega1 of U and B which we denote by G(omega1) (U, B) . This game is like the ordinary Ehrenfeucht-Fraisse-game of L(omegaomega) except that there are omega1 moves. It is clear that G(omega1) (U, B) is determined if U and B are of cardinality less-than-or-equal-to aleph1. We prove the following r esults: Theorem 1. If V = L, then there are models U and B of cardinal ity aleph2 such that the game G(omega1) (U, B) is nondetermined. Theor em 2. If it is consistent that there is a measurable cardinal, then it is consistent that G(omega1) (U, B) is determined for all U and B of cardinality less-than-or-equal-to aleph2. Theorem 3. For any kappa gre ater-than-or-equal-to aleph3 there are U and B of cardinality kappa su ch that the game G(omega1) (U, B) is nondetermined.