General topology meets model theory, on 𝔭 and 𝔱

Trans. Amer. Math. Soc.

2015

Abstract. This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(ℵ0, D) of ℵ0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by non-low theories. Assuming κ > ℵ0 is measurable, we construct a regular ultrafilter on λ ≥ 2 κ which is flexible but not good, and which moreover has large lcf(ℵ0) but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving non-low does not imply maximal. Since flexible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, ok need not imply good, addressing a problem from Dow 1985. Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for D regular on κ and M a model of an unstable theory, M κ /D is not (2 κ ) + -saturated; and for M a model of a non-simple theory and λ = λ <λ , M λ /D is not λ ++ -saturated. In the third part of the paper, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on SOP2. We prove that if D is a κ-complete ultrafilter on κ, any ultrapower of a sufficiently saturated model of linear order will have no (κ, κ)-cuts, and that if D is also normal, it will have a (κ + , κ + )-cut. We apply this to prove that for any n < ω, assuming the existence of n measurable cardinals below λ, there is a regular ultrafilter D on λ such that any D-ultrapower of a model of linear order will have n alternations of cuts, as defined below. Moreover, D will λ + -saturate all stable theories but will not (2 κ ) + -saturate any unstable theory, where κ is the smallest measurable cardinal used in the construction.

Section: Description Of Resultsmentioning

confidence: 96%

Constructing regular ultrafilters from a model-theoretic point of view

Trans. Amer. Math. Soc.

2015

“…Then s is a cofinality spectrum problem with exponentiation, and so Theorem 2.10 applies: p s = t s . 9 An independent proof that 1.1(6) will be satisfied on some pair, assuming only that the c.s.p.…”

Section: On Bounded Arithmeticmentioning

confidence: 99%

Model-theoretic applications of cofinality spectrum problems

2017

Isr. J. Math.

Self Cite

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...

“…Keisler's order is defined in §6. Further background on Keisler's order and saturation of ultrapowers appears in [23] and in the introduction to [24]. Earlier sources are [11], [12].…”

mentioning

confidence: 99%

Keisler’s order has infinitely many classes

2018

Isr. J. Math.

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We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler's order. Thus Keisler's order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler's order is a central notion of the model theory of the 60s and 70s which compares first-order theories (and implicitly ultrafilters) according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a very natural class of theories, the n-free k-hypergraphs studied by Hrushovski. This complexity reflects the difficulty of amalgamation and appears orthogonal to forking.A significant challenge to our understanding of unstable theories in general, and simple theories in particular, has been the apparent intractability of the problem of Keisler's order. Defined in 1967 [11], Keisler's order uses saturation of ultrapowers to compare the complexity of any two complete, countable first-order theories (basic model-theoretic objects of study). The recent articles [23], [25] explain the fundamental interest of the classification program arising from: Question 0.1 (Keisler 1967). Determine the structure of Keisler's order.The structure of Keisler's order on the stable theories was settled in Chapter VI of the second author's 1978 book [28]. For a long time, there was little progress on the unstable case; [19] gives some history. This was due both to the difficulty of constructing ultrafilters and the state of our understanding of unstable theories. In the last few years, there has been very substantial progress in this area , Malliaris and Shelah [19]-[24]).Notably, Keisler's order was thought to be finite, with perhaps four classes, whose identities were suggested in [28] Problem VI.0.1. After [20] and [24], the number was at least five. In the present paper, we leverage the ZFC theorems of [24] to prove, nearly fifty years after Keisler introduced the order, that:Main Theorem (Theorem 6.6 below, in ZFC). There is an infinite strictly descending sequence of simple unstable theories in Keisler's order. Thus, Keisler's order is infinite and not a well order.The proof builds on a breakthrough of [24], the isolation of 'explicit simplicity' and the corresponding construction of a family of so-called optimal ultrafilters. In some sense, such ultrafilters are to simple theories as Keisler's good ultrafilters