Theories with EF-Equivalent Non-Isomorphic Models

by Shelah. [Sh:897]
Tbilisi Math J, 2008
Our ``large scale'' aim is to characterize the first order T (at least the countable ones) such that: for every ordinal alpha there lambda,M_1,M_2 such that M_1,M_2 are non-isomorphic models of T of cardinality lambda which are EF_{alpha, lambda}-equivalent. We expect that as in the main gap ([Sh:c,XII]) we get a strong dichotomy, so in the non-structure side we have more, better example, and in the structure side we have a parallel of [Sh:c,XIII]. We presently prove the consistency of the non-structure side for T which is aleph_0-independent (= not strongly dependent) or just not strongly stable, even for PC (T_1,T) and more for unstable T (see [Sh:c,VII] or [Sh:h]) and infinite linear order I .

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