Abstract
We give a new characterization of SOP (the strict order property) in terms of the behaviour of formulas in any model of the theory as opposed to having to look at the behaviour of indiscernible sequences inside saturated ones. We refine a theorem of Shelah, namely a theory has OP (the order property) if and only if it has IP (the independence property) or SOP, in several ways by characterizing various notions in functional analytic style. We point out some connections between dividing lines in first order theories and subclasses of Baire 1 functions, and give new characterizations of some classes and new classes of first order theories.
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Notes
In this article, when we refer to Shelah’s theorem, we mean this theorem.
We should be more careful with this assertion; if the variables y are just dummy variables that play no role, we can not recover A from X. This result is true for the space of full types, but not necessarily for just the \({{\tilde{\phi }}}\)-types. However, for the sake of simplicity we continue to write \(A\subseteq C(X)\).
Although these notions seems very restrictive and unnatural, they are very useful for proving the main theorem of this section, i.e., Theorem 2.6 below. Note that the notion \(\phi \)-n-type is completely different from the notion \(\phi \)-type we defined earlier.
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Acknowledgements
I am very much indebted to Professor John T. Baldwin for his kindness and his helpful comments. I want to thank Pierre Simon for his interest in reading a preliminary version of this article and for his comments. I thank the anonymous referee for his/her detailed suggestions and corrections; they helped to improve significantly the exposition of this paper. I would like to thank the Institute for Basic Sciences (IPM), Tehran, Iran. Research partially supported by IPM Grant No. 99030117.
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Khanaki, K. Dividing lines in unstable theories and subclasses of Baire 1 functions. Arch. Math. Logic 61, 977–993 (2022). https://doi.org/10.1007/s00153-022-00816-8
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DOI: https://doi.org/10.1007/s00153-022-00816-8