# Sh:416

- Mekler, A. H., Shelah, S., & Väänänen, J. A. (1993).
*The Ehrenfeucht-Fraïssé-game of length \omega_1*. Trans. Amer. Math. Soc.,**339**(2), 567–580. arXiv: math/9305204 DOI: 10.2307/2154287 MR: 1191613 -
Abstract:

Let ({\mathcal A}) and ({\mathcal B}) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraı̈ssé-game of length \omega_1 of {\mathcal A} and {\mathcal B} which we denote by G_{\omega_1}({\mathcal A},{\mathcal B}). This game is like the ordinary Ehrenfeucht-Fraı̈ssé-game of L_{\omega\omega} except that there are \omega_1 moves. It is clear that G_{\omega_1}({\mathcal A},{\mathcal B}) is determined if \mathcal A and {\mathcal B} are of cardinality \leq\aleph_1. We prove the following results:Theorem A: If V=L, then there are models \mathcal A and \mathcal B of cardinality \aleph_2 such that the game G_{\omega_1}({\mathcal A},{\mathcal B}) is non-determined.

Theorem B: If it is consistent that there is a measurable cardinal, then it is consistent that G_{\omega_1}({\mathcal A},{\mathcal B}) is determined for all \mathcal A and \mathcal B of cardinality \le\aleph_2.

Theorem C: For any \kappa\geq\aleph_3 there are \mathcal A and \mathcal B of cardinality \kappa such that the game G_{\omega_1}({\mathcal A},{\mathcal B}) is non-determined.

- published version (15p)

Bib entry

@article{Sh:416, author = {Mekler, Alan H. and Shelah, Saharon and V{\"a}{\"a}n{\"a}nen, Jouko A.}, title = {{The Ehrenfeucht-Fra\"iss\'e-game of length $\omega_1$}}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the American Mathematical Society}, volume = {339}, number = {2}, year = {1993}, pages = {567--580}, issn = {0002-9947}, mrnumber = {1191613}, mrclass = {03C55 (03E05 03E35 03E55 90D44)}, doi = {10.2307/2154287}, note = {\href{https://arxiv.org/abs/math/9305204}{arXiv: math/9305204}}, arxiv_number = {math/9305204} }