Model-Theoretic Properties of Ultrafilters Built by Independent Families of Functions

Trans. Amer. Math. Soc.

2015

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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: Summary Theoremsmentioning

confidence: 92%

“…(This paper and its sequel [15] contain at least three distinct proofs of that fact, of independent interest.) The numbering of results follows that in §7.…”

Section: Definition 15 (Ok Ultrafilters)mentioning

confidence: 93%

Constructing regular ultrafilters from a model-theoretic point of view

Trans. Amer. Math. Soc.

2015

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“…For more on flexibility, see Malliaris [13] and recent work of Malliaris and Shelah [15]- [16], where it is shown that flexible is consistently weaker than good.…”

Section: )mentioning

confidence: 99%

A dividing line within simple unstable theories

Advances in Mathematics

2013

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106

Abstract. We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is µ < λ ≤ 2 µ , we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or of the random graph,The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr 1 , generalizing the fact that whenever B is a set of parameters in some sufficiently saturated model of the random graph, |B| = λ and µ < λ ≤ 2 µ , then there is a set A with |A| = µ so that any nonalgebraic p ∈ S(B) is finitely realized in A. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of "excellence," a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of moral ultrafilters on Boolean algebras. We prove a so-called "separation of variables" result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set-theoretic stage, building an excellent filter, followed by a more model-theoretic stage: building so-called moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from first-order formulas, in certain Boolean algebras.

“…There is a property of filters, called flexibility, which is detected by any theory that is nonlow (some formula k-divides with respect to arbitrarily large k). Keisler's order imposes a hierarchy on the structure/ randomness phenomenon from Discussion 5: The "structured" (in some sense, rich) theories, those with linear (strict) order, are Keislermaximum whereas the "purest" random theory, that of the Rado graph, is Keisler- (26,(34)(35)(36). These theorems are not prerequisites for the current proofs, so we refer the interested reader to the introduction of ref.…”

Section: Cardinal Invariants Of the Continuummentioning

confidence: 99%

General topology meets model theory, on 𝔭 and 𝔱

Proc. Natl. Acad. Sci. U.S.A.

2013

Cantor proved in 1874 [Cantor G (1874) J Reine Angew Math 77: [258][259][260][261][262]] that the continuum is uncountable, and Hilbert's first problem asks whether it is the smallest uncountable cardinal. A program arose to study cardinal invariants of the continuum, which measure the size of the continuum in various ways. By Gödel [Gödel K (1939) Proc Natl Acad Sci USA 25(4):220-224] and Cohen [Cohen P (1963) Proc Natl Acad Sci USA 50(6):1143-1148], Hilbert's first problem is independent of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Much work both before and since has been done on inequalities between these cardinal invariants, but some basic questions have remained open despite Cohen's introduction of forcing. The oldest and perhaps most famous of these is whether "p = t," which was proved in a special case by Rothberger [Rothberger F (1948) I n this paper we present our solution to two long-standing, and a priori unrelated, questions: the question from set theory/ general topology of whether p = t, the oldest problem on cardinal invariants of the continuum, and the question from model theory of whether SOP 2 is maximal in Keisler's order. Before motivating and precisely stating these questions, we note there were two big surprises in this work: first, the connection of the two questions and, second, that the question of whether "p < t" can be decided in ZFC (Zermelo-Fraenkel set theory with the axiom of choice).There have been few connections between general topology and model theory, and these were exclusively in model theory: a key example is Morley's use of ideas from general topology such as the Cantor-Bendixson derivative in proving his 1965 categoricity theorem (1), the cornerstone of modern model theory. Here we have applications in the other direction: by our proofs below, model theoretic methods solve an old problem in general topology. It seems too early to tell what further interactions will arise from these methods, but it is a good sign that Claim 1 below will be generalized to PðNÞ=fin; see Discussion 21. Full details of work presented here are available in ref. 2.