Invariants related to the tree property
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INVARIANTS RELATED TO THE TREE PROPERTY BY
Nicholas Ramsey Department of Mathematics, University of California, Berkeley Berkeley, CA 94720-3840, USA e-mail: [email protected]
ABSTRACT
We consider cardinal invariants related to Shelah’s model-theoretic tree properties and the relations that obtain between them. From strong colorings, we construct theories T with κcdt (T ) > κsct (T ) + κinp (T ). We show that these invariants have distinct structural consequences, by investigating their effect on the decay of saturation in ultrapowers. This answers some questions of Shelah.
1. Introduction One of the fundamental discoveries in stability theory is that stability is local: a theory is stable if and only if no formula has the order property. Among the stable theories, one can obtain a measure of complexity by associating to each theory T its stability spectrum, namely, the class of cardinals λ such that T is stable in λ. A classification of stability spectra was given by Shelah in [She90, Chapter 3]. Part of this analysis amounts to showing that stable theories do not have the tree property and, consequently, that forking satisfies local character. But a crucial component of that work was studying the approximations to the tree property which can exist in stable theories and what structural consequences they have. These approximations were measured by a cardinal invariant of the theory called κ(T ), and Shelah’s stability spectrum theorem gives an explicit description of the cardinals in which a given theory T is stable in terms of the cardinality of the set of types in finitely many Received May 25, 2018 and in revised form September 26, 2019
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Isr. J. Math.
variables over the empty set and κ(T ). Shelah used the definition of κ(T ) as a template for quantifying the global approximations to other tree properties in introducing the invariants κcdt (T ), κsct (T ), and κinp (T ) (see Definition 2.1 below) which bound approximations to the tree property (TP), the tree property of the first kind (TP1 ), and the tree property of the second kind (TP2 ), respectively. Eventually, the local condition that a theory does not have the tree property (simplicity), and the global condition that κ(T ) = κcdt (T ) = ℵ0 (supersimplicity) proved to mark substantial dividing lines. These invariants provide a coarse measure of the complexity of the theory, providing a “quantitative” description of the patterns that can arise among forking formulas. They are likely to continue to play a role in the development of a structure theory for tame classes of non-simple theories. Motivated by some questions from [She90], we explore which relationships known to hold between the local properties TP, TP1 , and TP2 also hold for the global invariants κcdt (T ), κsct (T ), and κinp (T ). In short, we are pursuing the following analogy: This continues the work done in [CR16], where, with Artem local TP global κcdt
TP1 κsct
TP2 κinp
Chernikov, we considered a global analogue of the following theorem of Shelah: Theorem ([She90, II
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