The Ehrenfeucht-Fraïssé-game of length $\omega _ 1$

Abstract:

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