# Sh:401

• Shelah, S. (2004). Characterizing an \aleph_\epsilon-saturated model of superstable NDOP theories by its \mathbb L_{\infty,\aleph_\epsilon}-theory. Israel J. Math., 140, 61–111.
• Abstract:
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\times {}^\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\Rightarrow\alpha<\omega_1] and [\eta\in {}^\omega 2 \backslash S\Rightarrow\alpha<\omega]\}. Hence, it seems to me we should try L_{\infty,\aleph_\epsilon}(d.q.) (essentially, in {\mathfrak 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_{{\mathcal 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.
• Version 2005-12-20_10 (61p) published version (51p)
Bib entry
@article{Sh:401,
author = {Shelah, Saharon},
title = {{Characterizing an $\aleph_\epsilon$-saturated model of superstable NDOP theories by its $\mathbb L_{\infty,\aleph_\epsilon}$-theory}},
journal = {Israel J. Math.},
fjournal = {Israel Journal of Mathematics},
volume = {140},
year = {2004},
pages = {61--111},
issn = {0021-2172},
mrnumber = {2054839},
mrclass = {03C45 (03C50)},
doi = {10.1007/BF02786627},
note = {\href{https://arxiv.org/abs/math/9609215}{arXiv: math/9609215}},
arxiv_number = {math/9609215}
}