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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 $$PC(T_1,T)$$.

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