By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, , is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if , 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 , holds, then there are ,ℬ ⊨ T of power ℵ₃ such that is non-determined.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-5,
author = {Tapani Hyttinen and Saharon Shelah and Jouko Vaananen},
title = {More on the Ehrenfeucht-Fraisse game of length o1},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {79-96},
zbl = {1013.03047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-5}
}
Tapani Hyttinen; Saharon Shelah; Jouko Vaananen. More on the Ehrenfeucht-Fraisse game of length ω₁. Fundamenta Mathematicae, Tome 173 (2002) pp. 79-96. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-5/