Sheaves and normal submodels

1977 ◽  
Vol 42 (2) ◽  
pp. 241-250 ◽  
Author(s):  
Richard Mansfield
Keyword(s):  
Model Theory ◽  
Heyting Algebra ◽  
Global Section ◽  
Principal Result ◽  
Traditional Model ◽  
Boolean Valued Model ◽  
Definition Of ◽  
Interesting Generalization ◽  
Horn Sentences

Ellerman, Comer, and Macintyre have all observed that sheaves are an interesting generalization of models and are deserving of model theoretic attention. Scott has pointed out that sheaves are Heyting algebra valued models. The reverse does not hold however since almost no genuine Boolean valued model is a sheaf.In §1 we shall review the definition of a sheaf and prove a theorem about Boolean valued models using the sheaf construction. In §2 we shall be concerned with the set of sentences preserved by global sections. Our principal result is that global section sentences are also normal submodel sentences. (We define as a normal submodel of if is a submodel of and every point of B − A can be moved by an automorphism of which fixes each point of A.) In §3 we prove that every normal submodel sentence is the negation of a disjunction of Horn sentences and that the set of normal submodel sentences is r.e. but not recursive. §3 involves only traditional model theory and can be read independently of the first two sections.

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.

scholarly journals First Order Languages: Further Syntax and Semantics

2011 ◽  
Vol 19 (3) ◽  
pp. 179-192 ◽  
Author(s):  
Marco Caminati
Keyword(s):  
Model Theory ◽  
Atomic Formula ◽  
Order Model ◽  
Logical Implication ◽  
First Order ◽  
Definition Of ◽  
Theory Interpretation

First Order Languages: Further Syntax and SemanticsThird of a series of articles laying down the bases for classical first order model theory. Interpretation of a language in a universe set. Evaluation of a term in a universe. Truth evaluation of an atomic formula. Reassigning the value of a symbol in a given interpretation. Syntax and semantics of a non atomic formula are then defined concurrently (this point is explained in [16], 4.2.1). As a consequence, the evaluation of any w.f.f. string and the relation of logical implication are introduced. Depth of a formula. Definition of satisfaction and entailment (aka entailment or logical implication) relations, see [18] III.3.2 and III.4.1 respectively.

scholarly journals Sensible or Outdated? Gender and Opinions of Tenure Criteria in Canada

2019 ◽  
Vol 49 (2) ◽  
pp. 1-16 ◽  
Author(s):  
Jennifer Dengate ◽  
Annemieke Farenhorst ◽  
Tracey Peter
Keyword(s):  
Academic Success ◽  
Gender Bias ◽  
Traditional Model ◽  
Gender Similarities ◽  
Research Scholarship ◽  
The Status ◽  
Allocation Of Time ◽  
Definition Of Success ◽  
Definition Of ◽  
The University

The university reward structure has traditionally placed greater value on individual research excellence for tenure and promotion, influencing faculty’s allocation of time and definition of worthwhile labour. We find gender differences in Canadian natural sciences and engineering faculty’s opinions of the traditional criteria for measuring academic success that are consistent with an implicit gender bias devaluing service and teamwork. Most women recommend significant changes to the traditional model and its foundation, while a substantial minority of men support the status quo. However, this comparative qualitative analysis finds more cross-gender similarities than differences, as most men also want a more modern definition of success, perceiving the traditional model to be disproportionately supportive of one type of narrow research scholarship that does not align with the realities of most faculty’s efforts. Thus, this study suggests a discrepancy between traditional success criteria and faculty’s understanding of worthwhile labour.

Tarski

Truth ◽  
2011 ◽  
Author(s):  
Alexis G. Burgess ◽  
John P. Burgess
Keyword(s):  
Mathematical Logic ◽  
Model Theory ◽  
The State ◽  
Truth Predicate ◽  
Object Language ◽  
French And English ◽  
Direct Definition ◽  
Definition Of ◽  
Basic Features ◽  
Self Reference

This chapter offers a simplified account of the most basic features of Alfred Tarski's model theory. Tarski foresaw important applications for a notion of truth in mathematics, but also saw that mathematicians were suspicious of that notion, and rightly so given the state of understanding of it circa 1930. In a series of papers in Polish, German, French, and English from the 1930s onward, Tarski attempted to rehabilitate the notion for use in mathematics, and his efforts had by the 1950s resulted in the creation of a branch of mathematical logic known as model theory. The chapter first considers Tarski's notion of truth, which he calls “semantic” truth, before discussing his views on object language and metalanguage, recursive versus direct definition of the truth predicate, and self-reference.

On the “Third Definition” of the Topology on the Spectrum of a C*-Algebra

1971 ◽  
Vol 23 (3) ◽  
pp. 445-450 ◽  
Author(s):  
L. Terrell Gardner
Keyword(s):  
Hilbert Space ◽  
Quotient Space ◽  
Principal Result ◽  
Irreducible Representations ◽  
C Algebra ◽  
Fell Topology ◽  
The Third ◽  
Definition Of ◽  
Image Position

0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the formwhere the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.

Filter Constructions in Boolean Valued Model Theory

1972 ◽  
Vol 18 (13-15) ◽  
pp. 193-200
Author(s):  
Alan M. Shorb
Keyword(s):  
Model Theory ◽  
Boolean Valued Model

MINIMAL AND STRANGE ATTRACTORS

1996 ◽  
Vol 06 (06) ◽  
pp. 1177-1183 ◽  
Author(s):  
A. GORODETSKI ◽  
Yu. ILYASHENKO
Keyword(s):  
Dynamical System ◽  
General Concept ◽  
Principal Result ◽  
Strange Attractors ◽  
Mild Form ◽  
Attracting Set ◽  
Probabilistic Character ◽  
A Chain ◽  
The One ◽  
Definition Of

A general concept going back to Kolmogorov claims that if a dynamical system has a complicated attracting set then its behavior has not a deterministic, but rather probabilistic character. This concept was not formalized up to now. Even the definition of attractor has a lot of different versions. This paper presents an attempt to give some definitions and results formalizing this heuristic ideas. It contains a definition of a minimal attractor, modifying the one given in Ilyashenko [1991]. The actual minimality of the attractor is discussed. The principal result is the Triple Choice Theorem. It claims that the existence of a strange minimal attractor implies some mild form of chaos for the map itself or for a nearby one. The program of further investigation is proposed as a chain of problems at the end of the paper.

To the definition of operators , which are performed IN Kaluzhnin’s graph-schems with parafractal characteristics

2020 ◽  
pp. 98-104
Author(s):  
V.M. Simonov
Keyword(s):  
Standard Procedure ◽  
Principal Result ◽  
Operator Equations ◽  
Definition Of ◽  
System Of Operator Equations

The regular formula of operator, which is performed in given Kaluzhnin’s graph-scheme with parafractal characteristics, can be definded by two procedures. The standard procedure is based on the solution of system of operator equations, which is given birth by this graph-scheme. The modified procedure is based on the solution of several lesser-scale systems of operator equations, which are given birth by parafractals. The modified procedure is simpler than the standard one, but it is not evident identity of the results of both procedures. The principal result of this article is the theorem about this identity.

The Future Neolithic

Early Farmers ◽  
2014 ◽  
Author(s):  
John Robb
Keyword(s):  
Principal Result ◽  
Specific Kind ◽  
Material World ◽  
Quiet Revolution ◽  
The Future ◽  
Definition Of ◽  
The Relationship

As for Shakespeare, every generation gets the Neolithic it deserves. This chapter discusses emerging views of what the Neolithic is and how to study it, with the thesis that recently there has been a quiet revolution in how we understand the Neolithic. A broad change in how Neolithic specialists understand the relationship between science and the humanities is envisioned, with the principal result that a new interpretive vocabulary, including a definition of the Neolithic, has arisen. This is illustrated with regard to changing understandings in the study of animals, plants, landscapes, things and monuments; for example, rather than being culture written on the material world, or material worlds determining culture, the practices of Neolithic life defined participation in a specific kind of historic process structured by these relations. The effects of changing perspectives are shown in, for example, multi-scalar approaches to both the origins and the end of the Neolithic.

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