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Preprint Number 1503

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1503. M. Malliaris and S. Shelah
A separation theorem for simple theories

Submission date: 22 October 2018


This paper builds model-theoretic tools to detect changes in complexity among the simple theories. We develop a generalization of dividing, called shearing, which depends on a so-called context c. This leads to defining c-superstability, a syntactical notion, which includes supersimplicity as a special case. We prove a separation theorem showing that for any countable context c and any two theories T_1, T_2 such that T_1 is c-superstable and T_2 is c-unsuperstable, and for arbitrarily large μ, it is possible to build models of any theory interpreting both T_1 and T_2 whose restriction to τ(T_1) is μ-saturated and whose restriction to τ(T_2) is not ℵ_1-saturated. (This suggests “c-superstable” is really a dividing line.) The proof uses generalized Ehrenfeucht-Mostowski models, and along the way, we clarify the use of these techniques to realize certain types while omitting others. In some sense, shearing allows us to study the interaction of complexity coming from the usual notion of dividing in simple theories and the more combinatorial complexity detected by the general definition. This work is inspired by our recent progress on Keisler's order, but does not use ultrafilters, rather aiming to build up the internal model theory of these classes.

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Full text arXiv 1810.09604: pdf, ps.

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