Results 1 
5 of
5
A dividing line within simple unstable theories.
, 2013
"... 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 ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
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, M λ /D is λ +saturated but (ii) if Th(N) is not simple or not low then N λ /D is not λ +saturated. The nonsaturation 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 socalled “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 settheoretic stage, building an excellent filter, followed by a more modeltheoretic stage: building socalled moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from firstorder formulas, in certain Boolean algebras.
CONSTRUCTING REGULAR ULTRAFILTERS FROM A MODELTHEORETIC POINT OF VIEW
"... Abstract. This paper contributes to the settheoretic 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, ..."
Abstract

Cited by 6 (6 self)
 Add to MetaCart
(Show Context)
Abstract. This paper contributes to the settheoretic 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 nonlow 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 nonlow does not imply maximal. Since flexible is precisely OK, this also shows that (c) from a settheoretic 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 nonsimple theories. More precisely, for D regular on κ and M a
Saturating the random graph with an independent family of small range
, 2012
"... re vi si on:2 ..."
(Show Context)
EXISTENCE OF OPTIMAL ULTRAFILTERS AND THE FUNDAMENTAL COMPLEXITY OF SIMPLE THEORIES
"... re vi si on:2 ..."
(Show Context)