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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Alan Mekler, Saharon Shelah and Jouko Väänänen
Journal: Trans. Amer. Math. Soc. 339 (1993), 567-580
MSC: Primary 03C55; Secondary 03E05, 03E35, 03E55, 90D44
MathSciNet review: 1191613
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1191613-1
PII: S 0002-9947(1993)1191613-1
Article copyright: © Copyright 1993 American Mathematical Society