The stability spectrum for classes of atomic models

by Baldwin and Shelah. [BlSh:959]
J Math Logic, 2012
We prove two results on the stability spectrum for L_{omega_1, omega} . Here S^m_i(M) denotes an appropriate notion (at (+ 1 962) or 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 boldmath K with |S^m_i(M)| > |M|^{beth_alpha (|T|)} . Then for every lambda >= |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 mbox {boldmath K} such that lambda_alpha = |M_{alpha}| >= beth_alpha and |S^m_{i}(M_alpha)| > lambda_alpha . Then for any mu with mu^{aleph_0}> mu, mbox {boldmath 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~ label {treeindis}, we expound the construction of tree indiscernibles for sentences of L_{omega_1, omega} .


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