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.

Type reducing correspondences and well-orderings: Frege's and Zermelo's constructions re-examined

1995 ◽  
Vol 60 (1) ◽  
pp. 209-221 ◽  
Author(s):  
J. L. Bell
Keyword(s):  
Choice Function ◽  
Axiom Of Choice ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Cardinal Numbers ◽  
The Axiom Of Choice

A key idea in both Frege's development of arithmetic in the Grundlagen [7] and Zermelo's 1904 proof [10] of the well-ordering theorem is that of a “type reducing” correspondence between second-level and first-level entities. In Frege's construction, the correspondence obtains between concept and number, in Zermelo's (through the axiom of choice), between set and member. In this paper, a formulation is given and a detailed investigation undertaken of a system ℱ of many-sorted first-order logic (first outlined in the Appendix to [6]) in which this notion of type reducing correspondence is accorded a central role and which enables Frege's and Zermelo's constructions to be presented in such a way as to reveal their essential similarity. By adapting Bourbaki's version of Zermelo's proof of the well-ordering theorem, we show that, within ℱ, any correspondence c between second-level entities (here called concepts) and first-level ones (here called objects) induces a well-ordering relation W (c) in a canonical manner. We shall see that, when c is the “Fregean” correspondence between concepts and cardinal numbers, W (c) is (the well-ordering of) the ordinal ω + 1, and when c is a “Zermelian” choice function on concepts, W (c) is a well-ordering of the universal concept embracing all objects.In ℱ an important role is played by the notion of extension of a concept. To each concept X we assume there is assigned an object e(X) in such a way that, for any concepts X, Y satisfying a certain predicate E, we have e (X) = e (Y) iff the same objects fall under X and Y.

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.

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.

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.

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.

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.

Gödel, Kurt (1906–78)

2018 ◽  
Author(s):  
John W. Dawson
Keyword(s):  
Twentieth Century ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Mathematical Platonism ◽  
The Axiom Of Choice ◽  
Fundamental Theorems ◽  
The Continuum

The greatest logician of the twentieth century, Gödel is renowned for his advocacy of mathematical Platonism and for three fundamental theorems in logic: the completeness of first-order logic; the incompleteness of formalized arithmetic; and the consistency of the axiom of choice and the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory.

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.

The Hanf number for complete Lω1,ω-sentences (without GCH)

1974 ◽  
Vol 39 (3) ◽  
pp. 575-578 ◽  
Author(s):  
James E. Baumgartner
Keyword(s):  
Continuum Hypothesis ◽  
Order Logic ◽  
First Order Logic ◽  
Crucial Point ◽  
Basic Approach ◽  
First Order ◽  
Large Cardinality ◽  
Countable Ordinal ◽  
Generalized Continuum ◽  
Hanf Number

The Hanf number for sentences of a language L is defined to be the least cardinal κ with the property that for any sentence φ of L, if φ has a model of power ≥ κ then φ has models of arbitrarily large cardinality. We shall be interested in the language Lω1,ω (see [3]), which is obtained by adding to the formation rules for first-order logic the rule that the conjunction of countably many formulas is also a formula.Lopez-Escobar proved [4] that the Hanf number for sentences of Lω1,ω is ⊐ω1, where the cardinals ⊐α are defined recursively by ⊐0 = ℵ0 and ⊐α = Σ{2⊐β: β < α} for all cardinals α > 0. Here ω1 denotes the least uncountable ordinal.A sentence of Lω1,ω is complete if all its models satisfy the same Lω1,ω-sentences. In [5], Malitz proved that the Hanf number for complete sentences of Lω1,ω is also ⊐ω1, but his proof required the generalized continuum hypothesis (GCH). The purpose of this paper is to give a proof that does not require GCH.More precisely, we will prove the following:Theorem 1. For any countable ordinal α, there is a complete Lω1,ω-sentence σαwhich has models of power ⊐α but no models of higher cardinality.Our basic approach is identical with Malitz's. We simply use a different combinatorial fact at the crucial point.

scholarly journals Notes on Quasiminimality and Excellence

2004 ◽  
Vol 10 (3) ◽  
pp. 334-366 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Special Kind ◽  
Order Logic ◽  
First Order Logic ◽  
Structure Theory ◽  
Mathematical Structures ◽  
First Order ◽  
Elementary Class ◽  
Abstract Elementary Class

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

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