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Preprint Number 852
852. M. Malliaris and S. Shelah
Model-theoretic applications of cofinality spectrum problems
Submission date: 28 March 2015.
We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has cofinality ≥ λ and the underlying order has no (κ, κ)-cuts for regular κ < λ. Second, assuming instances of GCH, we prove that SOP_2 characterizes maximality in the interpretability order ⊴^*, settling a prior conjecture and proving that SOP_2 is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP_2, proving that NSOP_2 can be characterized by the existence of few inconsistent higher formulas. In the course of the paper, we show that p_s = t_s in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.
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