More on the Ehrenfeucht–Fraisse game of length
Tom 175 / 2002
Streszczenie
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.
