The Ehrenfeucht-Fraisse-game of length omega_1 [article]

Alan H. Mekler, Saharon Shelah, Jouko Väänänen
1993 arXiv   pre-print
Let (A) and (B) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraisse-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_1. We prove the following results: Theorem A: If V=L, then there are models A and B of cardinality aleph_2 such that the
more » ... e G_omega_1(A,B) is non-determined. Theorem B: If it is consistent that there is a measurable cardinal, then it is consistent that G_omega_1(A,B) is determined for all A and B of cardinality <= aleph_2. Theorem C: For any kappa >= aleph_3 there are A and B of cardinality kappa such that the game G_omega_1(A,B) is non-determined.
arXiv:math/9305204v1 fatcat:pcjkyiee6jaejapovvti2zgefu