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## Fundamenta Mathematicae

2002 | 175 | 1 | 79-96
Tytuł artykułu

### More on the Ehrenfeucht-Fraisse game of length ω₁

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By results of [9] there are models 𝔄 and 𝔅 for which the Ehrenfeucht-Fraïssé game of length ω₁, $EFG_{ω₁}(𝔄,𝔅)$, 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 $EFG_{ω₁}(𝔄,𝔅)$ is determined for all models 𝔄 and 𝔅 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 𝓐,ℬ ⊨ T of power ℵ₃ such that $EFG_{ω₁}(𝓐,ℬ)$ is non-determined.
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Tom
Numer
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79-96
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wydano
2002
Twórcy
autor
• Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), 00014 University of Helsinki, Finland
autor
• Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
• Deparment of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A.
autor
• Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), 00014 University of Helsinki, Finland
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