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.

• Version 1993-08-29_10 (22p) published version (15p)

@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}
}