Model-theoretic applications of cofinality spectrum problems

by Malliaris and Shelah. [MiSh:1051]

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 lambda-saturated iff it has cofinality >= lambda and the underlying order has no (kappa, kappa)-gaps for regular kappa < lambda . We also answer a question about balanced pairs of models

of PA. Second, assuming instances of GCH, we prove that SOP_2 characterizes maximality in the interpretability order triangleleft^*, 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 so-called higher formulas. In the course of the paper, we show that {p}_{mathbf {s}} = {t}_{mathbf {s}} in emph {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. *2015.02.12 The new note from Feb on the delayed lemma make no sense:- the finite satisfiability ensure the formulas are compatible. Look back at the old.. The second theorem which is delayed (union of few pairwise non-contradictory) - a debt, look again


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