The metamathematics of model theory: Discovering language in action

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.

Download Full-text

SPACES OF TYPES IN POSITIVE MODEL THEORY

Journal of Symbolic Logic ◽

10.1017/jsl.2018.84 ◽

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.

Download Full-text

Pure-projective modules and positive constructibility

Journal of Symbolic Logic ◽

10.2307/2586526 ◽

2000 ◽

Vol 65 (1) ◽

pp. 103-110 ◽

Cited By ~ 1

Author(s):

T. G. Kucera ◽

Ph. Rothmaler

Keyword(s):

Model Theory ◽

Projective Module ◽

Quantifier Elimination ◽

Combine Model ◽

Projective Modules ◽

Module Theory ◽

First Order ◽

The One ◽

Countable Models

In modules many ‘positive’ versions of model-theoretic concepts turn out to be equivalent to concepts known in classical module theory—by ‘positive’ we mean that instead of allowing arbitrary first-order formulas in the model-theoretic definitions only positive primitive formulas are taken into consideration. (This feature is due to Baur's quantifier elimination for modules, cf. [Pr], however we will not make explicit use of it here.) Often this allows one to combine model-theoretic methods with algebraic ones. One instance of this is the result proved in [Rot1] (see also [Rot2]) that the Mittag-Leffler modules are exactly the positively atomic modules. This paper is parallel to the one just mentioned in that it is proved here, Theorem 3.1, that the pure-projective modules are exactly the positively constructible modules. The following parallel facts from module theory and from model theory led us to this result: every pure-projective module is Mittag-Leffler and the converse is true for countable (in fact even countably generated) modules, cf. [RG]; every constructible model is atomic and the converse is true for countable models, cf. [Pi].

Download Full-text

On countable fractions from an elementary class

Journal of Symbolic Logic ◽

10.2307/2275713 ◽

1994 ◽

Vol 59 (4) ◽

pp. 1410-1413

Author(s):

C. J. Ash

Keyword(s):

Model Theory ◽

Relation Symbol ◽

First Order ◽

Elementary Class ◽

Free Variables ◽

Order Language ◽

Skolem Theorem ◽

Elementary Result ◽

Countable Models

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.

Download Full-text

A Definability Theorem for First Order Logic

BRICS Report Series ◽

10.7146/brics.v4i3.18782 ◽

1997 ◽

Vol 4 (3) ◽

Cited By ~ 1

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. 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 automorphisms of M. The same is of course true for subsets of M" 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 of L. 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 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.

Download Full-text

Sheaves and Boolean valued model theory

Journal of Symbolic Logic ◽

10.2307/2273725 ◽

1979 ◽

Vol 44 (2) ◽

pp. 153-183 ◽

Cited By ~ 17

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.

Download Full-text

Boolean-valued structures

10.1093/oso/9780198790396.003.0013 ◽

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.

Download Full-text

Weakly atomic-compact relational structures

Journal of Symbolic Logic ◽

10.2307/2271522 ◽

1971 ◽

Vol 36 (1) ◽

pp. 129-140 ◽

Cited By ~ 4

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]).

Download Full-text

Model Theory of Epimorphisms

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-083-6 ◽

1974 ◽

Vol 17 (4) ◽

pp. 471-477 ◽

Cited By ~ 6

Author(s):

Paul D. Bacsich

Keyword(s):

Model Theory ◽

Order Theory ◽

First Order ◽

First Order Theory

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .

Download Full-text

Some model theory for almost real closed fields

Journal of Symbolic Logic ◽

10.2307/2275808 ◽

1996 ◽

Vol 61 (4) ◽

pp. 1121-1152 ◽

Cited By ~ 5

Author(s):

Françoise Delon ◽

Rafel Farré

Keyword(s):

Model Theory ◽

Residue Field ◽

Elementary Equivalence ◽

First Order ◽

Valuation Rings ◽

Closed Fields ◽

Theory Of Fields ◽

Model Theory Of Fields ◽

Elementary Inclusion ◽

Henselian Valuation

AbstractWe study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

Download Full-text