The Ehrenfeucht-Fra{\"\i}ss{\'e}-game of length $\omega_1$

by Mekler and Shelah and Vaananen. [MShV:416]
Transactions American Math Soc, 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-Fra{i}sse-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 game 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.

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