Model-theoretic applications of cofinality spectrum problems

Abstract

Abstract. 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)-gaps 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 N SOP 2 , proving for instance that N SOP 2 can be characterized in terms of few inconsistent higher formulas. In the course of the paper, we show that ps = ts 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.In a recent paper [8] we connected and solved two a priori unrelated open questions: the question from model theory of whether SOP 2 is maximal in Keisler's order, and the question from set theory/general topology of whether p = t. This work was described in the research announcement [9] and the commentary [11]. In order to prove these theorems, we introduced a general framework called cofinality spectrum problems, reviewed in §1 below.In the present paper, we develop and apply cofinality spectrum problems to a range of problems in model theory, primarily on Peano arithmetic and around the strong tree property SOP 2 , also called the 2-strong order property. We prove the following theorems:Theorem (Theorem 4.7). Let N be a model of Peano arithmetic, or just bounded PA, and λ an uncountable cardinal. If the reduct of N to the language of order has cofinality > κ and no (κ, κ)-cuts for all κ < λ, then N is λ-saturated. Theorem (Theorem 2.10). Let s be a cofinality spectrum problem which is closed under exponentiation. Then p s = t s .As explained in §1, Theorem 2.10 complements a main theorem of [8], which showed that t s ≤ p s for any cofinality spectrum problem s. As a consequence, we are able to characterize t s in terms of the first symmetric cut. However, as we show in §5, the local versions of these cardinals, p s,a , t s,a need not coincide unless the underlying model is saturated.We then turn to the strong tree property SOP 2 . A major result of [8] was that SOP 2 suffices for being maximal in Keisler's order . It was not proved to be a necessary condition, but we conjectured there that SOP 2 characterizes maximality in Keisler's order. The difficulty in addressing this question may be in building ultrafilters. However, in the present paper, for a related open problem, we give a complete answer:Theorem (Theorem 6.14, under relevant instances of GCH). T is * -maximal if and only if it has SOP 2 . The ordering* refines Keisler's order, but is defined not in terms of ultrapowers but rather in terms of interpretability. Theorem 6.14 answers a very interesting question going back to Džamonja and Shel...

Cofinality spectrum theorems in model theory, set theory, and general topology

“…Our work here also gives a new characterization of Keisler's good ultrafilters, Theorem 10.25 quoted below. Cofinality spectrum problems include other central examples, such as Peano arithmetic, as developed further in our paper [31].…”

Divide and Conquer: Dividing Lines and Universality

“…Theorem 5.3. (Assuming an instance of GCH) (one direction (Džamonja and Shelah, 2004a;Shelah and Usvyatsov, 2008b) and the other (Malliaris and Shelah, 2017)) T is ⊲ * -maximal iff T is SOP 2 (so SOP 2 is a robust pre-dividing line! ).…”

“…The conjecture is that any simple theory has λ-PC-exact saturation for all singular λ, or at least for all strong limit cardinals λ. There is evidence (by work in progress, see Malliaris and Shelah, 2017) that "singular PC-exact saturation" may turn out to be a good test problem for a dividing line between simple and NSOP 2 . • The quite old question about the order ≤ SP , related to Theorem 4.2, see Shelah and Ulrich (2018).…”

More on tree properties

“…Later in [12], type-counting criteria for TP 1 (equivalently for SOP 2 ) analogous to the type-counting results of [17] are suggested. In [15], another typecounting criteria for SOP 2 is suggested. Now in this section, we supply more refined criteria for SOP 2 , which are analogous to those for TP in [1].…”