Characterizing an $\aleph_\epsilon $-saturated model of superstable NDOP theories by its $\mathbb L_{\infty,\aleph_\epsilon}$-theory

by Shelah. [Sh:401]
Israel J Math, 2004
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 {}^omega 2,E_n)_{n< omega} where (alpha, eta)E_n(beta, nu) =: eta |n= nu |n and for S subseteq

{}^omega 2 we define M_S = M| {(alpha, eta):[eta in S => alpha < omega_1] and [eta in {}^omega 2 backslash

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 Leftrightarrow M cong N for aleph_epsilon-saturated models of a superstable countable (first order) theory T without dop.

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