scholarly journals SPACES OF TYPES IN POSITIVE MODEL THEORY

2019 ◽  
Vol 84 (02) ◽  
pp. 833-848
Author(s):  
LEVON HAYKAZYAN
Keyword(s):  
Model Theory ◽  
Stone Space ◽  
Distributive Lattices ◽  
Stone Duality ◽  
Order Model ◽  
First Order ◽  
Countable Models

AbstractWe introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

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.

THE FIRST-ORDER THEORY OF BRANCH GROUPS

2016 ◽  
Vol 102 (1) ◽  
pp. 150-158 ◽  
Author(s):  
JOHN S. WILSON
Keyword(s):  
Model Theory ◽  
Order Theory ◽  
Order Model ◽  
First Order ◽  
First Order Theory

It is shown that for many branch groups $G$ the action on the ambient tree can be interpreted in $G$, in the sense of first-order model theory.

Infinitary analogs of theorems from first order model theory

1971 ◽  
Vol 36 (2) ◽  
pp. 216-228 ◽  
Author(s):  
Jerome Malitz
Keyword(s):  
Model Theory ◽  
Expressive Power ◽  
Direct Products ◽  
Order Model ◽  
Relation Symbol ◽  
First Order ◽  
Identity Symbol

The material presented here belongs to the model theory of the Lκ, λ languages. Our results are either infinitary analogs of important theorems in finitary model theory, or else show that such analogs do not exist.For example, it is well known that whenever i, and , have the same true Lω, ω sentences (i.e., are elementarily equivalent) for i = 1, 2, then the cardinal sums 1 + 2 and + have the same true Lω, ω sentences, and the direct products 1 · 2 and · have the same true Lω, ω sentences [3]. We show that this is true when ‘Lω, ω’ is replaced by ‘Lκ, λ’ if and only if κ is strongly inaccessible. For Lω1, ω this settles a question, posed by Lopez-Escobar [7].In §3 we give a complete description of the expressive power of those sentences of Lκ, λ in which the identity symbol is the only relation symbol which occurs. This extends a result by Hanf [4].

On nonstandard models in higher order logic

1984 ◽  
Vol 49 (1) ◽  
pp. 204-219
Author(s):  
Christian Hort ◽  
Horst Osswald
Keyword(s):  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
Elementary Embedding ◽  
Simple Type ◽  
Order Model ◽  
Higher Order Logic ◽  
First Order ◽  
Simple Type Theory ◽  
Nonstandard Models

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.

scholarly journals Compactness and Independence in Non First Order Frameworks

2005 ◽  
Vol 11 (1) ◽  
pp. 28-50 ◽  
Author(s):  
Itay Ben-Yaacov
Keyword(s):  
Model Theory ◽  
Finite Cover ◽  
Order Model ◽  
Simple Theories ◽  
First Order ◽  
The Cost

AbstractThis communication deals with positive model theory, a non first order model theoretic setting which preserves compactness at the cost of giving up negation. Positive model theory deals transparently with hyperimaginaries, and accommodates various analytic structures which defy direct first order treatment. We describe the development of simplicity theory in this setting, and an application to the lovely pairs of models of simple theories without the weak non finite cover property.

scholarly journals First-order model theory of free projective planes

2021 ◽  
Vol 172 (2) ◽  
pp. 102888
Author(s):  
Tapani Hyttinen ◽  
Gianluca Paolini
Keyword(s):  
Model Theory ◽  
Projective Planes ◽  
Order Model ◽  
First Order
Sign in / Sign up

Export Citation Format

Share Document