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Theories with Ehrenfeucht-Fraïssé equivalent non-isomorphic models. (English) Zbl 1158.03315
Summary: Our “long term and large scale” aim is to characterize the first order theories \(T\) (at least the countable ones) such that for every ordinal \(\alpha\) there are \(\lambda\), \(M_1\), \(M_2\) such that \(M_1\) and \(M_2\) are non-isomorphic models of \(T\) of cardinality \(\lambda\) which are \(EF^+_{\alpha,\lambda}\)-equivalent. We expect that as in the main gap [S. Shelah, Classification theory and the number of non-isomorphic models. 2nd rev. ed. Amsterdam: North-Holland (1990; Zbl 0713.03013), XII], we get a strong dichotomy, i.e., on the non-structure side we have stronger, better examples, and on the structure side we have an analogue of [loc. cit., XIII]. We presently prove the consistency of the non-structure side for \(T\) which is \(\aleph_0\)-independent (= not strongly dependent), even for \(PC(T_1,T)\).

03C55 Set-theoretic model theory
03C68 Other classical first-order model theory
03E40 Other aspects of forcing and Boolean-valued models