More on the Ehrenfeucht–Fraisse game of length $\omega _1$

Tom 175 / 2002

Tapani Hyttinen, Saharon Shelah, Jouko Vaananen Fundamenta Mathematicae 175 (2002), 79-96 MSC: 03C55, 03C75, 03C45. DOI: 10.4064/fm175-1-5


By results of [9] there are models ${\frak A}$ and ${\frak B}$ for which the Ehrenfeucht–Fraïssé game of length $\omega _1$, ${\rm EFG}_{\omega _1}({\frak A},{\frak B})$, is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality $\le \aleph _2$. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and ${\rm EFG}_{\omega _1}({\frak A},{\frak B})$ is determined for all models ${\frak A}$ and ${\frak B}$ of cardinality $\aleph _2$” is that of a weakly compact cardinal. On the other hand, we show that if $2^{\aleph _0}<2^{\aleph _{3}}$, $T$ is a countable complete first order theory, and one of

(i) $T$ is unstable,

(ii) $T$ is superstable with DOP or OTOP,

(iii) $T$ is stable and unsuperstable and $2^{\aleph _0}\le \aleph _{3}$,

holds, then there are ${\cal A},{\cal B}\mathrel |\mathrel {\mkern -3mu}=T$ of power $\aleph _{3}$ such that ${\rm EFG}_{\omega _{1}}({\cal A},{\cal B})$ is non-determined.


  • Tapani HyttinenDepartment of Mathematics
    P.O. Box 4 (Yliopistonkatu 5)
    00014 University of Helsinki, Finland
  • Saharon ShelahEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Jerusalem 91904, Israel
    Deparment of Mathematics
    Rutgers University
    New Brunswick, NJ 08903, U.S.A.
  • Jouko VaananenDepartment of Mathematics
    P.O. Box 4 (Yliopistonkatu 5)
    00014 University of Helsinki, Finland

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