# 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. arXiv: math/9609215 DOI: 10.1007/BF02786627 MR: 2054839 -
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. - 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}, doi = {10.1007/BF02786627}, mrclass = {03C45 (03C50)}, mrnumber = {2054839}, doi = {10.1007/BF02786627}, note = {\href{https://arxiv.org/abs/math/9609215}{arXiv: math/9609215}}, arxiv_number = {math/9609215} }