The Ehrenfeucht-Fraïssé-game of length 𝜔₁

Abstract

Let A \mathfrak {A} and B \mathfrak {B} be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length ω 1 {\omega _1} of A \mathfrak {A} and B \mathfrak {B} which we denote by G ω 1 ( A , B ) {\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B}) . This game is like the ordinary Ehrenfeucht-Fraïssé-game of L ω ω {L_{\omega \omega }} except that there are ω 1 {\omega _1} moves. It is clear that G ω 1 ( A , B ) {\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B}) is determined if A \mathfrak {A} and B \mathfrak {B} are of cardinality ≤ ℵ 1 \leq {\aleph _1} . We prove the following results: Theorem 1. If V = L V = L , then there are models A \mathfrak {A} and B \mathfrak {B} of cardinality ℵ 2 {\aleph _2} such that the game G ω 1 ( A , B ) {\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B}) is nondetermined . Theorem 2. If it is consistent that there is a measurable cardinal, then it is consistent that G ω 1 ( A , B ) {\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B}) is determined for all A \mathfrak {A} and B \mathfrak {B} of cardinality ≤ ℵ 2 \leq {\aleph _2} . Theorem 3. For any κ ≥ ℵ 3 \kappa \geq {\aleph _3} there are A \mathfrak {A} and B \mathfrak {B} of cardinality κ \kappa such that the game G ω 1 ( A , B ) {\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B}) is nondetermined.