Undecidability of satisfiability of expansions of FO [ < ] over words with a  FO [ + ]-definable set

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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Preliminaries

10.23943/princeton/9780691161686.003.0002 ◽

2017 ◽

Author(s):

Ehud Hrushovski ◽

François Loeser

Keyword(s):

Valued Fields ◽

Definable Set ◽

Valued Field ◽

Galois Coverings ◽

Definable Type ◽

Definable Sets ◽

Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

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TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

Journal of Symbolic Logic ◽

10.1017/jsl.2016.45 ◽

2017 ◽

Vol 82 (1) ◽

pp. 347-358 ◽

Cited By ~ 4

Author(s):

PABLO CUBIDES KOVACSICS ◽

LUCK DARNIÈRE ◽

EVA LEENKNEGT

Keyword(s):

Algebraic Geometry ◽

Open Subset ◽

Cell Decomposition ◽

Real Algebraic Geometry ◽

Dimension Theory ◽

Definable Set ◽

Image Position ◽

Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

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Finitely axiomatizable ω-categorical theories and the Mazoyer hypothesis

Journal of Symbolic Logic ◽

10.2178/jsl/1120224723 ◽

2005 ◽

Vol 70 (2) ◽

pp. 460-472 ◽

Cited By ~ 1

Author(s):

David Lippel

Keyword(s):

Order Property ◽

Simple Theories ◽

Definable Set

AbstractLet F be the class of complete, finitely axiomatizable ω-categorical theories. It is not known whether there are simple theories in F. We prove three results of the form: if T ∈ F has a sufficently well-behaved definable set J, then T is not simple. (In one case, we actually prove that T has the strict order property.) All of our arguments assume that the definable set J satisfies the Mazoyer hypothesis, which controls how an element of J can be algebraic over a subset of the model. For every known example in F, there is a definable set satisfying the Mazoyer hypothesis.

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BookTubers as a Networked Knowledge Community

Emerging Pedagogies in the Networked Knowledge Society - Advances in Knowledge Acquisition, Transfer, and Management ◽

10.4018/978-1-4666-4757-2.ch004 ◽

2013 ◽

pp. 87-99 ◽

Cited By ~ 3

Author(s):

Karen Sorensen ◽

Andrew Mara

Keyword(s):

Social Media ◽

New Media ◽

Cultural Capital ◽

Knowledge Society ◽

Young Adult Fiction ◽

Discourse Community ◽

Knowledge Communities ◽

Knowledge Community ◽

Definable Set ◽

The Relationship

In order to understand the relationship between Networked Knowledge Communities (NKCs) and the Networked Knowledge Society (NKS), the chapter authors conduct a genre analysis of a self-titled New Media genre called BookTube. BookTube is a NKC made up of YouTube content creators who use this particular social media channel to celebrate and discuss books, especially young-adult fiction. By examining how BookTube adheres to discourse community features—shared rules, genres, hierarchies, and values—the contours of this particular NKC become clearer. Stylistic patterns, the roles of authors, and author cultural capital all get negotiated within a discernible and definable set of practices that relate participants to other NKCs and the broader NKS. Furthermore, by relating these discourse features to other latent educational possibilities of the NKS, the authors explore how BookTube might be usefully implemented to model NKC practices in more traditional f2f educational settings.

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GANZSTELLENSÄTZE IN THEORIES OF VALUED FIELDS

Journal of Mathematical Logic ◽

10.1142/s0219061308000695 ◽

2008 ◽

Vol 08 (01) ◽

pp. 1-22 ◽

Cited By ~ 1

Author(s):

DEIRDRE HASKELL ◽

YOAV YAFFE

Keyword(s):

Valuation Ring ◽

Uniform Method ◽

Algebraic Characterization ◽

Valued Fields ◽

Definable Set ◽

Valued Field ◽

Complete Theories ◽

The Given ◽

Relevant Class

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields.

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Types omitted in uncountable models of arithmetic

Journal of Symbolic Logic ◽

10.2307/2272157 ◽

1975 ◽

Vol 40 (3) ◽

pp. 317-320 ◽

Cited By ~ 3

Author(s):

Julia F. Knight

Keyword(s):

Completeness Theorem ◽

Elementary Extension ◽

Ramsey's Theorem ◽

Definable Set ◽

Ramsey’S Theorem ◽

Hanf Number ◽

Definable Sets ◽

Omitting Types ◽

Infinite Set ◽

Models Of Arithmetic

In [4] it is shown that if the structure omits a type Σ, and Σ is complete with respect to Th(), then there is a proper elementary extension of which omits Σ. This result is extended in the present paper. It is shown that Th() has models omitting Σ in all infinite powers.A type is a countable set of formulas with just the variable ν occurring free. A structure is said to omit the type Σ if no element of satisfies all of the formulas of Σ. A type Σ, in the same language as a theory T, is said to be complete with respect to T if (1) T ∪ Σ is consistent, and (2) for every formula φ(ν) of the language of T (with just ν free), either φ or ¬φ is in Σ.The proof of the result of this paper resembles Morley's proof [5] that the Hanf number for omitting types is . It is shown that there is a model of Th() which omits Σ and contains an infinite set of indiscernibles. Where Morley used the Erdös-Rado generalization of Ramsey's theorem, a definable version of the ordinary Ramsey's theorem is used here.The “omitting types” version of the ω-completeness theorem ([1], [3], [6]) is used, as it was in Morley's proof and in [4]. In [4], satisfaction of the hypotheses of the ω-completeness theorem followed from the fact that, in , any infinite, definable set can be split into two infinite, definable sets.

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Sets of theorems with short proofs

Journal of Symbolic Logic ◽

10.2307/2272636 ◽

1974 ◽

Vol 39 (2) ◽

pp. 235-242 ◽

Cited By ~ 13

Author(s):

Daniel Richardson

Keyword(s):

Proof Complexity ◽

Predicate Calculus ◽

Definable Set ◽

Semantic Tableaux ◽

Skolem Functions ◽

Definition Of

R. Parikh has shown that in the predicate calculus without function symbols it is possible to decide whether or not a given formula A is provable in a Hilbert-style proof of k lines. He has also shown that for any formula A(x1, …, xn) of arithmetic in which addition and multiplication are represented by three-place predicates, {(a1, …, an): A(a1, …, an) is provable from the axioms of arithmetic in k lines} is definable in (N,′, +,0). See [2].In §1 of this paper it is shown that if S is a definable set of n-tuples in (N,′, +, 0), then there is a formula A(x1, …, xn) and a number k so that (a1 …, an) ∈ S if and only if A(a1 …, an) can be proved in a proof of complexity k from the axioms of arithmetic. The result does not depend on which formalization of arithmetic is used or which (reasonable) measure of proof complexity. In particular, this gives a converse to Parikh's result.In §II, a measure of proof complexity is given which is based on counting the quantifier steps in semantic tableaux. The idea behind this measure is that A is k provable if in the attempt to construct a counterexample one meets insurmountable difficulties in the definition of the appropriate Skolem functions over sets of cardinality ≤ k. A method is given for deciding whether or not a sentence A in the full predicate calculus is k provable. The question for the full predicate calculus and Hilbert-style systems of proof remains open.

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