# Sh:959

• Baldwin, J. T., & Shelah, S. (2012). The stability spectrum for classes of atomic models. J. Math. Log., 12(1), 1250001, 19.
• Abstract:
We prove two results on the stability spectrum for L_{\omega_1,\omega}. Here S^m_i(M) denotes an appropriate notion ({\rm at}(+ 1 962) or {\rm mod}) of Stone space of m-types over M. Theorem A. Suppose that for some positive integer m and for every \alpha< \delta(T), there is an M \in \mathbf K with |S^m_i(M)| > |M|^{\beth_\alpha(|T|)}. Then for every \lambda \geq |T|, there is an M with |S^m_i(M)| > |M|. Theorem B. Suppose that for every \alpha<\delta(T), there is M_\alpha \in \mathbf K such that \lambda_\alpha = |M_{\alpha}| \geq \beth_\alpha and |S^m_{i}(M_\alpha)| > \lambda_\alpha. Then for any \mu with \mu^{\aleph_0}>\mu \mathbf K is not i-stable in \mu. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study. In the Section [treeindis], we expound the construction of tree indiscernibles for sentences of L_{\omega_1,\omega}.
• Current version: 2012-02-21_11 (19p) published version (19p)
Bib entry
@article{Sh:959,
author = {Baldwin, John T. and Shelah, Saharon},
title = {{The stability spectrum for classes of atomic models}},
journal = {J. Math. Log.},
fjournal = {Journal of Mathematical Logic},
volume = {12},
number = {1},
year = {2012},
pages = {1250001, 19},
issn = {0219-0613},
mrnumber = {2950191},
mrclass = {03C45 (03C75)},
doi = {10.1142/S0219061312500018}
}