More on the Ehrenfeucht–Fraisse game of length $\omega _1$
Volume 175 / 2002
Abstract
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.
