The Ehrenfeucht-Fraisse-Game of Length ω 1
Alan H. Mekler, Saharon Shelah, Jouko Väänánen · Transactions of the American Mathematical Society · 1993
Let A and B be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length ω1 of A and B which we denote by Gω1 (A, B). This game is like the ordinary Ehrenfeucht-Fraïssé-game of Lωω except that there are ω1 moves. It is clear that Gω1 (A, B) is determined if A and B are of cardinality ≤ ℵ1. We prove the following results: 1 Theorem 1 If V=L, then there are models A and B of cardinality ℵ2 such that the game Gω1 (A, B) is non-determined. Theorem 2 If it is consistent that there is a measurable cardinal, then it is consistent that Gω1 (A, B) is determined for all A and B of