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Cofinality spectrum theorems in model theory, set theory and general topology, arXiv eprint 1208.5424
, 2012
"... Abstract. We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t, the oldest problem on cardinal invariants of the continuum. We do so ..."
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Abstract. We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t, the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using modeltheoretic methods. By a cofinality spectrum problem s we essentially mean the data of a pair of models M M1 which code sufficient set theory, possibly in an expanded language, along with a distinguished set of formulas ∆s which define linear orders in M1. Let ts, the “treetops ” of s, be the smallest regular cardinal λ such that one of a set of derived trees in M1 has a strictly increasing λsequence with no upper bound. Let C(s, ts) be the set of pairs of regular cardinals (κ1, κ2) such that κ1 ≤ κ2 < ts and some ∆sdefinable linear order contains a (κ1, κ2)cut. We prove that for any cofinality spectrum problem s, C(s, ts) = ∅. Using this theorem and framework we prove first, that SOP2 is maximal in Keisler’s order; second, that p = t; and third, that any regular ultrafilter D on λ for which “ts> λ, ” or what is equivalent, such that (ω,<)λ/D contains no (κ, κ)cuts for κ = cf(κ) ≤ λ, is λ+good. We obtain several consequences, notably existence of a minimum Keisler class among the nonsimple theories.
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 ..."
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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.
Modeltheoretic properties of ultrafilters built by independent families of functions.” math.LO/1208.2579
"... Abstract. Via two short proofs and three constructions, we show how to increase the modeltheoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus satura ..."
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Abstract. Via two short proofs and three constructions, we show how to increase the modeltheoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck ” in the inductive construction of a regular ultrafilter on λ (i.e. a point after which all antichains of P(λ)/D have cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter D on λ so that any ultrafilter extending D fails to λ+saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal κ, we construct a regular ultrafilter on λ> κ which is λflexible but not κ++good, improving our previous answer to a question raised in Dow 1975. Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that lcf(ℵ0) may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory. Our methods advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. 1.
Saturating the random graph with an independent family of small range
, 2012
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EXISTENCE OF OPTIMAL ULTRAFILTERS AND THE FUNDAMENTAL COMPLEXITY OF SIMPLE THEORIES
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