Regular ultrafilters and finite square principles

Kennedy, Juliette; Shelah, Saharon; Väänánen, Jouko

doi:10.2178/jsl/1230396748

Cited by 6 publications

(12 citation statements)

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“…(The second author's book [20].VI contains a first proof of this result.) The argument uses a combinatorial principle from [7] against the definition of the tree property.…”

Section: Description Of Resultsmentioning

confidence: 99%

Constructing regular ultrafilters from a model-theoretic point of view

Malliaris

Shelah

2015

Trans. Amer. Math. Soc.

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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.

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“…(The second author's book [20].VI contains a first proof of this result.) The argument uses a combinatorial principle from [7] against the definition of the tree property.…”

Section: Description Of Resultsmentioning

confidence: 99%

Constructing regular ultrafilters from a model-theoretic point of view

Malliaris

Shelah

2015

Trans. Amer. Math. Soc.

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“…The notion (λ, μ)-regular ultrafilter has proven particularly useful in Model Theory, e. g., [6,7,12,34,43,48], Set Theory, e. g., [1,16,21,25,27,32,33,67], and even General Topology, e. g., [10,23,40,42,49,52,62]. See also, e. g., [5,13,17,39,55,56,58] for interdisciplinary and further applications.…”

Section: Introductionmentioning

confidence: 99%

More on regular and decomposable ultrafilters in ZFC

Lipparini¹

2010

Mathematical Logic Qtrly

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Abstract. We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them:(ii) If κ is regular then D is either (λ, κ)-regular, or (ω, κ ′ )-regular for every κ ′ < κ (Corollary 6.4).(iii) If either (1) λ is a strong limit cardinal and λ <λ < 2 κ , or (2) λ <λ < κ, then D is either λ-decomposable, or (λ ′ , κ)-regular for some λ ′ < λ (Theorem 6.5).(d) If λ is singular, D is (µ, cfλ)-regular and there are arbitrarily large ν < λ for which D is ν-decomposable then D is κ-decomposable for some κ with λ ≤ κ ≤ λ <µ (Theorem 5.1; actually, our result is stronger and involves a covering number).(e) D × D ′ is (λ, µ)-regular if and only if there is a ν such that D is (ν, µ)-regular and D ′ is (λ, ν ′ )-regular for all ν ′ < ν (Proposition 7.1).We also list some problems, and furnish applications to topological spaces and to extended logics (Corollaries 4.6 and 4.8).

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“…Discussion 6.5. In [15] we gave a result showing "decay of saturation" for non-simple theories using a combinatorial principle from [9]. It would be very useful if the construction just given could be extended to e.g.…”

Section: Distinct Parametersmentioning

confidence: 99%

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

Malliaris¹,

Shelah²

2014

J. symb. log.

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Abstract. Via two short proofs and three constructions, we show how to increase the model-theoretic 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 non-low 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 λ-flexible 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(ℵ 0 ) may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of pre-cuts 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.

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Regular ultrafilters and finite square principles

Cited by 6 publications

References 5 publications

Constructing regular ultrafilters from a model-theoretic point of view

Constructing regular ultrafilters from a model-theoretic point of view

More on regular and decomposable ultrafilters in ZFC

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

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