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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.
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, ..."
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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
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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