On the number of generators of cylindric algebras

1985 ◽  
Vol 50 (4) ◽  
pp. 865-873
Author(s):  
H. Andréka ◽  
I. Németi
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Single Element ◽  
Abstract Model ◽  
Finitely Generated ◽  
Cylindric Algebras ◽  
Abstract Model Theory ◽  
First Order ◽  
Cylindric Set Algebras

The theory of cylindric algebras (CA's) is the algebraic theory of first order logics. Several ideas about logic are easier to formulate in the frame of CA-theory. Such are e.g. some concepts of abstract model theory (cf. [1] and [10]–[12]) as well as ideas about relationships between several axiomatic theories of different similarity types (cf. [4] and [10]). In contrast with the relationship between Boolean algebras and classical propositional logic, CA's correspond not only to classical first order logic but also to several other ones. Hence CA-theoretic results contain more information than their counterparts in first order logic. For more about this see [1], [3], [5], [9], [10] and [12].Here we shall use the notation and concepts of the monographs Henkin-Monk-Tarski [7] and [8]. ω denotes the set of natural numbers. CAα denotes the class of all cylindric algebras of dimension α; by “a CAα” we shall understand an element of the class CAα. The class Dcα ⊆ CAα was defined in [7]. Note that Dcα = 0 for α ∈ ω. The classes Wsα, and Csα were defined in 1.1.1 of [8], p. 4. They are called the classes of all weak cylindric set algebras, regular cylindric set algebras and cylindric set algebras respectively. It is proved in [8] (I.7.13, I.1.9) that ⊆ CAα. (These inclusions are proper by 7.3.7, 1.4.3 and 1.5.3 of [8].)It was proved in 2.3.22 and 2.3.23 of [7] that every simple, finitely generated Dcα is generated by a single element. This is the algebraic counterpart of a property of first order logics (cf. 2.3.23 of [7]). The question arose: for which simple CAα's does “finitely generated” imply “generated by a single element” (see p. 291 and Problem 2.3 in [7]). In terms of abstract model theory this amounts to asking the question: For which logics does the property described in 2.3.23 of [7] hold? This property is roughly the following. In any maximal theory any finite set of concepts is definable in terms of a single concept. The connection with CA-theory is that maximal theories correspond to simple CA's (the elements of which are the concepts of the original logic) and definability corresponds to generation.

Barwise: Abstract Model Theory and Generalized Quantifiers

2004 ◽  
Vol 10 (1) ◽  
pp. 37-53 ◽  
Author(s):  
Jouko Väänänen
Keyword(s):  
Model Theory ◽  
Interpolation Property ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
Abstract Model ◽  
Famous Theorem ◽  
Abstract Model Theory ◽  
First Order ◽  
Skolem Theorem

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties areκ-compactness. Any set of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model. Löwenheim-Skolem Theorem down to κ. If a sentence of the logic has a model, it has a model of cardinality at most κ. Interpolation Property. If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languages Lκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory.

Inverse topological systems and compactness in abstract model theory

1986 ◽  
Vol 51 (3) ◽  
pp. 785-794 ◽  
Author(s):  
Daniele Mundici
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Clopen Subset ◽  
Abstract Model ◽  
Topological Properties ◽  
Abstract Model Theory ◽  
Abstract Logic ◽  
First Order ◽  
Topological System ◽  
Directed Set

AbstractGiven an abstract logic , generated by a set of quantifiers Qi, one can construct for each type τ a topological space Sτ, exactly as one constructs the Stone space for τ in first-order logic. Letting T be an arbitrary directed set of types, the set is an inverse topological system whose bonding mappings are naturally determined by the reduct operation on structures. We relate the compactness of to the topological properties of ST. For example, if I is countable then is compact iff for every τ each clopen subset of Sτ is of finite type and Sτ, is homeomorphic to limST, where T is the set of finite subtypes of τ. We finally apply our results to concrete logics.

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.

Model theory under the axiom of determinateness

1985 ◽  
Vol 50 (3) ◽  
pp. 773-780
Author(s):  
Mitchell Spector
Keyword(s):  
Model Theory ◽  
Axiom Of Choice ◽  
Order Logic ◽  
First Order Logic ◽  
Technical Innovation ◽  
First Order ◽  
Partition Relations ◽  
Hanf Number ◽  
The Axiom Of Choice

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

Some Aspects of Model Theory and Finite Structures

2002 ◽  
Vol 8 (3) ◽  
pp. 380-403 ◽  
Author(s):  
Eric Rosen
Keyword(s):  
Computer Science ◽  
Complexity Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Model ◽  
Finite Model Theory ◽  
First Order ◽  
Single Sentence ◽  
Finite Structures

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

On amalgamations of languages with Magidor-Malitz quantifiers

1979 ◽  
Vol 44 (4) ◽  
pp. 549-558
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Universal Algebra ◽  
Model Theory ◽  
Natural Extension ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Free Variables ◽  
Countably Compact

In this paper we indicate how compact languages containing the Magidor-Malitz quantifiers Qκn in different cardinalities can be amalgamated to yield more expressive, compact languages.The language Lκ<ω, originally introduced by Magidor and Malitz [9], is a natural extension of the language L(Q) introduced by Mostowski and investigated by Fuhrken [6], [7], Keisler [8] and Vaught [13]. Intuitively, Lκ<ω is first-order logic together with quantifiers Qκn (n ∈ ω) binding n free variables which express “there is a set X of cardinality κ such than any n distinct elements of X satisfy …”, or in other words, iff the relation on determined by φ contains an n-cube of cardinality κ. With these languages one can express a variety of combinatorial statements of the type considered by Erdös and his colleagues, as well as concepts in universal algebra which are beyond the scope of first-order logic. The model theory of Lκ<ω has been further developed by Badger [1], Magidor and Malitz [10] and Shelah [12].We refer to a language as being < κ compact if, given any set of sentences Σ of the language, if Σ is finitely satisfiable and ∣Σ∣ < κ, then Σ has a model. The phrase countably compact is used in place of <ℵ1 compact.

scholarly journals Undecidable relativizations of algebras of relations

1999 ◽  
Vol 64 (2) ◽  
pp. 747-760 ◽  
Author(s):  
Szabolcs Mikulás ◽  
Maarten Marx
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Similarity Type ◽  
First Order ◽  
Equational Theories ◽  
Algebras Of Relations ◽  
Guarded Fragment ◽  
Cylindric Set Algebras

AbstractIn this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

Logic in the 1930s: Type Theory and Model Theory

2013 ◽  
Vol 19 (4) ◽  
pp. 433-472 ◽  
Author(s):  
Georg Schiemer ◽  
Erich H. Reck
Keyword(s):  
Twentieth Century ◽  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Gradual Transformation ◽  
Standard Framework ◽  
Theory And Model ◽  
Made In

AbstractIn historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition (Carnap and Tarski) and two proponents of the first-order tradition (Gödel and Hilbert), we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory.

Decidability of cylindric set algebras of dimension two and first-order logic with two variables

1999 ◽  
Vol 64 (4) ◽  
pp. 1563-1572 ◽  
Author(s):  
Maarten Marx ◽  
Szabolcs Mikulás
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Universal Theory ◽  
Finite Model ◽  
Finite Model Property ◽  
First Order ◽  
Combinatorial Theorem ◽  
Finite Base ◽  
Dimension 2 ◽  
Cylindric Set Algebras

AbstractThe aim of this paper is to give a new proof for the decidability and finite model property of first-order logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse2). The new proof also shows the known results that the universal theory of Pse2 is decidable and that every finite Pse2 can be represented on a finite base. Since the class Cs2 of cylindric set algebras of dimension 2 forms a reduct of Pse2, these results extend to Cs2 as well.

AN APPLICATION OF FIRST-ORDER LOGIC TO THE STUDY OF RECOGNIZABLE LANGUAGES

2004 ◽  
Vol 14 (05n06) ◽  
pp. 785-799 ◽  
Author(s):  
PEDRO V. SILVA
Keyword(s):  
Finite Rank ◽  
Free Monoid ◽  
Order Logic ◽  
First Order Logic ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
First Order ◽  
Free Inverse Monoid

A variation of first-order logic with variables for exponents is developed to solve some problems in the setting of recognizable languages on the free monoid, accommodating operators such as product, bounded shuffle and reversion. Restricting the operators to powers and product, analogous results are obtained for recognizable languages of an arbitrary finitely generated monoid M, in particular for a free inverse monoid of finite rank. As a consequence, it is shown to be decidable whether or not a recognizable subset of M is pure or p-pure.

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