More on the Ehrenfeucht-Fraisse game of length ω₁

Tapani Hyttinen, Saharon Shelah, and Jouko Vaananen. "More on the Ehrenfeucht-Fraisse game of length ω₁." Fundamenta Mathematicae 175.1 (2002): 79-96. <http://eudml.org/doc/282644>.

@article{TapaniHyttinen2002,
abstract = {By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, $EFG_\{ω₁\}(,)$, 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 $EFG_\{ω₁\}(,)$ is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if $2^\{ℵ₀\} < 2^\{ℵ₃\}$, 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 $2^\{ℵ₀\} ≤ ℵ₃$, holds, then there are ,ℬ ⊨ T of power ℵ₃ such that $EFG_\{ω₁\}(,ℬ)$ is non-determined.},
author = {Tapani Hyttinen, Saharon Shelah, Jouko Vaananen},
journal = {Fundamenta Mathematicae},
keywords = {stability; Ehrenfeucht-Fraïssé game; consistency strength},
language = {eng},
number = {1},
pages = {79-96},
title = {More on the Ehrenfeucht-Fraisse game of length ω₁},
url = {http://eudml.org/doc/282644},
volume = {175},
year = {2002},
}

TY - JOUR
AU - Tapani Hyttinen
AU - Saharon Shelah
AU - Jouko Vaananen
TI - More on the Ehrenfeucht-Fraisse game of length ω₁
JO - Fundamenta Mathematicae
PY - 2002
VL - 175
IS - 1
SP - 79
EP - 96
AB - By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, $EFG_{ω₁}(,)$, 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 $EFG_{ω₁}(,)$ is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if $2^{ℵ₀} < 2^{ℵ₃}$, 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 $2^{ℵ₀} ≤ ℵ₃$, holds, then there are ,ℬ ⊨ T of power ℵ₃ such that $EFG_{ω₁}(,ℬ)$ is non-determined.
LA - eng
KW - stability; Ehrenfeucht-Fraïssé game; consistency strength
UR - http://eudml.org/doc/282644
ER -