Definable functions in Urysohn’s metric space

Goldbring, Isaac

doi:10.1215/ijm/1373636691

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…As such, the Urysohn sphere is often used as a fundamental example of the kind of structure well-suited for study in continuous logic. Previous work on the model theory of the Urysohn sphere can be found in [7], [8], and [15].…”

Section: Introductionmentioning

confidence: 99%

Model theoretic properties of the Urysohn sphere

Conant

Terry

2016

Annals of Pure and Applied Logic

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We characterize model theoretic properties of the Urysohn sphere as a metric structure in continuous logic. In particular, our first main result shows that the theory of the Urysohn sphere is SOPn for all n ≥ 3, but does not have the fully finite strong order property. Our second main result is a geometric characterization of dividing independence in the theory of the Urysohn sphere. We further show that this characterization satisfies the extension axiom, and so forking and dividing are the same for complete types. Our results require continuous analogs of several tools and notions in classification theory. While many of these results are undoubtedly known to researchers in the field, they have not previously appeared in publication. Therefore, we include a full exposition of these results for general continuous theories. project to us, and for guidance throughout the development of our results. We also thank Dave Marker for many helpful conversations. Continuous Model TheoryWe assume the reader is familiar with the basic setting of continuous logic and model theory for bounded metric structures. An in-depth introduction can be found in [3]. Throughout this section, T denotes a complete theory in continuous logic, and M is a sufficiently saturated monster model of T . The letters A, B, C, . . . denote sets, and we write A ⊂ M to mean A ⊆ M and M is χ(A) + -saturated, where χ(A) is the density character of A. We will useā,b,c, . . . to denote tuples of elements, and a, b, c, . . . to denote singletons. We use ℓ(ā) to denote the length of a tuple, which may be infinite.Recall that types in continuous logic consist of conditions of the form "ϕ(x) = 0", where ϕ(x) is some formula. Given ǫ > 0, we use "ϕ(x) ≤ ǫ" and "ϕ(x) ≥ ǫ" to denote, respectively, the conditions "ϕ(x) .− ǫ = 0" and "ǫ . − ϕ(x) = 0", where, given r, s ∈ [0, 1], r .− s = max{0, r −s}. We use "ϕ(x) = ǫ" to denote the condition "|ϕ(x) − ǫ| = 0".We will exclusively study metric structures of diameter ≤ 1. Therefore, throughout this paper, when carrying out calculations with distances we use addition truncated at 1. We also adopt the convention that sup ∅ = 0 and inf ∅ = 1.2.1. Classification Theory. In this section, we define the continuous analogs of several "dividing lines" in Shelah's classification hierarchy. The translation of these properties to continuous logic is an ongoing process. For example, stability is discussed in [3] and definitions of the independence property, the tree property of the second kind, and the strict order property can be found in [2]. Our focus will be on the strong order property, which was first defined for classical logic in [10]. Following the style of [2] and [3], we give syntactic definitions of various strong order properties, which are obtained from the definitions in [10] via a standard transfer of discrete connectives to continuous ones (e.g. max in place of conjunction).

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Section: Introductionmentioning

confidence: 99%

Model theoretic properties of the Urysohn sphere

Conant

Terry

2016

Annals of Pure and Applied Logic

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“…Continuous model theory in its current form is developed in the papers [BBHU] and [BU]. The papers [Go1], [Go2], [Go3] deal with definability questions in metric structures. Randomizations of models are treated in [AK], [Be], [BK], [EG], [GL], [Ke1], and [Ke2].…”

Section: Introductionmentioning

confidence: 99%

Definable closure in randomizations

Andrews

Goldbring

Keisler

2015

Annals of Pure and Applied Logic

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“…However, there has yet to be any mention of what the definable sets or functions are in this theory. In fact, there had yet to be any real study of definable functions in any metric structure until the paper [4] analyzed the definable functions in the Urysohn sphere.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we only study the definable linear operators on Hilbert spaces, for studying arbitrary definable functions seems a bit out of reach at the moment. As in [4], the key observation is the following: If M is a metric structure, A ⊆ M is a parameterset, and f : M → M is an A-definable function, then for every x ∈ M , we have f (x) ∈ dcl(Ax), where dcl stands for definable closure. Thus, in any theory where dcl is well-understood, one can begin to understand the definable functions.…”

Section: Introductionmentioning

confidence: 99%