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