Sheaves and Boolean valued model theory

1979 ◽  
Vol 44 (2) ◽  
pp. 153-183 ◽  
Author(s):  
George Loullis
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
Topological Spaces ◽  
Local Homeomorphism ◽  
First Order ◽  
Logical Operations ◽  
Boolean Valued Model ◽  
Work Done ◽  
Significant Model

In recent years model theorists have been studying various sheaf-theoretic notions as they apply to model theory. For quite a while however, a sheaf of structures was considered to be just a local homeomorphism between topological spaces such that each stalk Sx = p−1(x) is a model-theoretic structure and such that certain maps are continuous. Some of the model-theoretic work done with this notion of a sheaf of structures are the papers by Carson [2] and Macintyre [7]. Soon came the idea of considering a sheaf of structures not just as a collection of structures glued together in some continuous way, but rather as some sort of generalized structure. A significant model-theoretic study of sheaves in this new sense became possible only after the development of the theory of topoi. As F.W. Lawvere pointed out in [6], this represents the advance of mathematics (in our case the advance of model theory) from metaphysics to dialectics.A topos is the rather ingenious evolution of the notion of a Grothendieck topos [13]. It provides us with the idea that an object of a topos (e.g. the topos of sheaves over a topological space) may be thought of as a generalized set. Furthermore, all first-order logical operations have an interpretation in a topos, hence we may talk about generalized structures. Angus Macintyre suggested that some of his model-theoretic results about sheaves of structures may be understood better and perhaps simplified by doing model theory inside a topos of sheaves.

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

Many-valued logics of extended Gentzen style II

1970 ◽  
Vol 35 (4) ◽  
pp. 493-528 ◽  
Author(s):  
Moto-o Takahashi
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
Natural Extension ◽  
Preceding Paper ◽  
Considerable Extent ◽  
Continuous Logic ◽  
First Order ◽  
Natural Extensions ◽  
Fundamental Theorems ◽  
Order Continuous

In the monograph [1] of Chang and Keisler, a considerable extent of model theory of the first order continuous logic (that is, roughly speaking, many-valued logic with truth values from a topological space) is ingeniously developed without using any notion of provability.In this paper we shall define the notion of provability in continuous logic as well as the notion of matrix, which is a natural extension of one in finite-valued logic in [2], and develop the syntax and semantics of it mostly along the line in the preceding paper [2]. Fundamental theorems of model theory in continuous logic, which have been proved with purely model-theoretic proofs (i.e. those proofs which do not use any proof-theoretic notions) in [1], will be proved with proofs which are natural extensions of those in two-valued logic.

scholarly journals An elementary definability theorem for first order logic

1999 ◽  
Vol 64 (3) ◽  
pp. 1028-1036 ◽  
Author(s):  
C. Butz ◽  
I. Moerdijk
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Free Variables ◽  
Definable Subset ◽  
Boolean Valued Model

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.

The metamathematics of model theory: Discovering language in action

1981 ◽  
Vol 46 (3) ◽  
pp. 490-498
Author(s):  
Douglas E. Miller
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
First Order ◽  
Elementary Classes ◽  
Countable Models

AbstractWe discuss the problem of defining the collection of first-order elementary classes in terms of the natural topological space of countable models.

scholarly journals A Definability Theorem for First Order Logic

1997 ◽  
Vol 4 (3) ◽  
Author(s):  
Carsten Butz ◽  
Ieke Moerdijk
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Definable Subset ◽  
Language L ◽  
The One

In this paper, we will present a definability theorem for first order logic.<br />This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S M (i.e., a subset S = {a | M |= phi(a)} defined by some formula phi) is invariant under all<br />automorphisms of M. The same is of course true for subsets of M" defined<br />by formulas with n free variables.<br /> Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L.<br />Our presentation is entirely selfcontained, and only requires familiarity<br />with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning<br />Boolean valued models.<br />The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

On topological quotient maps preserved by pullbacks or products

1970 ◽  
Vol 67 (3) ◽  
pp. 553-558 ◽  
Author(s):  
B. J. Day ◽  
G. M. Kelly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Quotient Maps ◽  
Continuous Maps

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

scholarly journals Separation Axioms in Intuitionistic Fuzzy Topological Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Amit Kumar Singh ◽  
Rekha Srivastava
Keyword(s):  
Topological Space ◽  
Left Adjoint ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

In this paper we have studied separation axiomsTi,i=0,1,2in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functorsℬ:IF-Top→BF-Topand𝒟:BF-Top→IF-Topand observe that𝒟is left adjoint toℬ.

Uniformity on generalized topological spaces

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Dipankar Dey ◽  
Dhananjay Mandal ◽  
Manabendra Nath Mukherjee
Keyword(s):  
Topological Space ◽  
Present Article ◽  
Design Methodology ◽  
Original Work ◽  
Generalized Topology ◽  
Topological Spaces ◽  
Intermediate Structure ◽  
Content Type ◽  
Completely Regular ◽  
Generalized Topological Space

PurposeThe present article deals with the initiation and study of a uniformity like notion, captioned μ-uniformity, in the context of a generalized topological space.Design/methodology/approachThe existence of uniformity for a completely regular topological space is well-known, and the interrelation of this structure with a proximity is also well-studied. Using this idea, a structure on generalized topological space has been developed, to establish the same type of compatibility in the corresponding frameworks.FindingsIt is proved, among other things, that a μ-uniformity on a non-empty set X always induces a generalized topology on X, which is μ-completely regular too. In the last theorem of the paper, the authors develop a relation between μ-proximity and μ-uniformity by showing that every μ-uniformity generates a μ-proximity, both giving the same generalized topology on the underlying set.Originality/valueIt is an original work influenced by the previous works that have been done on generalized topological spaces. A kind of generalization has been done in this article, that has produced an intermediate structure to the already known generalized topological spaces.

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