Limits in the uniform ultrafilters

Baker, Joni; Kunen, Kenneth

doi:10.1090/s0002-9947-01-02843-4

Cited by 14 publications

(30 citation statements)

References 7 publications

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“…by requiring only that certain classes of functions have multiplicative refinements. An important example is the notion of OK, which appeared without a name in Keisler [4], was named and studied by Kunen [8] and investigated generally by Dow [3] and by Baker and Kunen [1]. We follow the definition from [3] 1.1.…”

Section: Background and Examplesmentioning

confidence: 99%

“…The statement that d has a multiplicative refinement is precisely the statement that there is, in fact, a common witness almost everywhere, in other words t ∈ d(σ) ∩ d(τ ) =⇒ t ∈ d(σ ∪ τ ). When this happens, we may choose at each index t an element c t such that σ ∈ (1). Then by Los' theorem and 1.7(1), t<λ c t will realize p in N .…”

Section: Definition 15 (Ok Ultrafilters)mentioning

confidence: 99%

“…Proof. (Sketch) (1) We first show that the following are equivalent: (i) any D-nonstandard integer projects E-a.e. to a D 1 -nonstandard integer, (ii) E is ℵ 1 -complete.…”

Section: Definition 15 (Ok Ultrafilters)mentioning

confidence: 99%

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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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Section: Background and Examplesmentioning

confidence: 99%

Section: Definition 15 (Ok Ultrafilters)mentioning

confidence: 99%

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Constructing regular ultrafilters from a model-theoretic point of view

Malliaris

Shelah

2015

Trans. Amer. Math. Soc.

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“…) Working in M, call an element of B + β decisive if it decides all formulas in Σ. Choose I a 11 As usual, we would like to replace au ℓ with (some smaller nonzero combination of elements from an earlier Boolean algebra and generators of the formal solution) as in 2.4 (1); but here we want to make sure, in addition, that the replacement is intelligible to Nu ℓ and also that the common intersection a remains nonzero in B. The simple acrobatics described carry this out.…”

Section: What Wementioning

confidence: 99%

“…Recall that the (theory of the) model-theoretic random graph is the model completion of the theory of graphs with a symmetric irrreflexive edge relation. That is, the language has a single binary relation R, 1 and there are axioms saying R is symmetric and irreflexive, that there are infinitely many elements, and for each finite m there is an axiom saying that given any two disjoint sets each with m elements, there is an x connected to every element in the first set and to no element in the second set.…”

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confidence: 99%

Some simple theories from a Boolean algebra point of view

Malliaris¹,

Shelah²

2021

Preprint

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We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories Tm reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories T n,k , which are the higher-order analogues of the triangle-free random graph. The proof involves building Boolean algebras and ultrafilters "by hand" to satisfy certain model theoretically meaningful chain conditions. This may be seen as advancing a line of work going back through Kunen's construction of good ultrafilters in ZFC using families of independent functions. We conclude with a theorem on flexible ultrafilters, and open questions.

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Ultrafilters Without Immediate Predecessors in Rudin–Frolik Order for Regulars

Jureczko

2022

Results Math

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The aim of this paper is to construct ultrafilters without immediate predecessors in the Rudin-Frolik order in $$\beta \kappa {\setminus } \kappa $$ β κ \ κ , where $$\kappa $$ κ is a regular cardinal. This generalizes the problem posed by Peter Simon more than 40 years ago.

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Limits in the uniform ultrafilters

Abstract: Abstract. Let u(κ) be the space of uniform ultrafilters on κ. If κ is regular, then there is an x ∈ u(κ) which is not an accumulation point of any subset of u(κ) of size κ or less. x is also good, in the sense of Keisler.

Cited by 14 publications

References 7 publications

Constructing regular ultrafilters from a model-theoretic point of view

Constructing regular ultrafilters from a model-theoretic point of view

Some simple theories from a Boolean algebra point of view

Ultrafilters Without Immediate Predecessors in Rudin–Frolik Order for Regulars

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