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A Hanf number for saturation and omission

John T. Baldwin, Saharon Shelah (2011)

Fundamenta Mathematicae

Suppose t = (T,T₁,p) is a triple of two countable theories T ⊆ T₁ in vocabularies τ ⊂ τ₁ and a τ₁-type p over the empty set. We show that the Hanf number for the property ’there is a model M₁ of T₁ which omits p, but M₁ ↾ τ is saturated’ is essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some distinctions between ’first order’ and ’second order quantification’. In particular, we show that if κ is uncountable,...

Amenability and Ramsey theory in the metric setting

Adriane Kaïchouh (2015)

Fundamenta Mathematicae

Moore [Fund. Math. 220 (2013)] characterizes the amenability of the automorphism groups of countable ultrahomogeneous structures by a Ramsey-type property. We extend this result to the automorphism groups of metric Fraïssé structures, which encompass all Polish groups. As an application, we prove that amenability is a G δ condition.

Categoricity of theories in Lκω , when κ is a measurable cardinal. Part 1

Saharon Shelah, Oren Kolman (1996)

Fundamenta Mathematicae

We assume a theory T in the logic L κ ω is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.

Finitary fibrations

Grzegorz Jarzembski (1989)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Levelled O-minimal structures.

David Marker, Chris Miller (1997)

Revista Matemática de la Universidad Complutense de Madrid

We introduce the notion of leveled structure and show that every structure elementarily equivalent to the real expo field expanded by all restricted analytic functions is leveled.

Metric abstract elementary classes with perturbations

Åsa Hirvonen, Tapani Hyttinen (2012)

Fundamenta Mathematicae

We define an abstract setting suitable for investigating perturbations of metric structures generalizing the notion of a metric abstract elementary class. We show how perturbation of Hilbert spaces with an automorphism and atomic Nakano spaces with bounded exponent fit into this framework, where the perturbations are built into the definition of the class being investigated. Further, assuming homogeneity and some other properties true in the example classes, we develop a notion of independence for...

Metric groups, unitary representations and continuous logic

Aleksander Ivanov (2021)

Communications in Mathematics

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find L ω 1 ω -axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

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