Characterizing aleph_epsilon-saturated models of superstable ndop theories by L_{infty, aleph_epsilon}-theory

After the main gap theorem was proved (see [Sh:c]), in discussion, Harrington expressed a desire for a finer structure - of finitary character (when we have a structure theorem at all). I point out that the logic L_{infty,aleph_0}(d.q.) (d.q. stands for dimension quantifier) does not suffice: e.g., for T=Th(lambda x 2^ω,E_n)_{n<omega} where (alpha,eta)E_n(beta,nu) =: eta|n=nu|n and for a subset S of 2^omega we define M_S = M | {(alpha,eta): [eta in S -> alpha<omega_1] and [eta in 2^ωbackslash S -> alpha<omega]}. Hence, it seems to me we should try L_{infty,aleph_epsilon}(d.q.) (essentially, in C we can quantify over sets which are included in the algebraic closure of finite sets), and Harrington accepts this interpretation. Here the conjecture is proved for aleph_epsilon-saturated models. I.e., the main theorem is M equiv_{L_{infty,aleph_epsilon}(d.q.)} N iff M cong N for aleph_epsilon--saturated models of a superstable countable (first order) theory T without dop.

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