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Applications of the Löwenheim–Skolem–Tarski Theorem to Problems of Completeness and Decidability
... If u is a first-order 9"-sentence, then abbreviate pz({U E dn(9') : U' t= u)), the fraction of members U of dn (9) for which 3' is a model of u, by pn(a). This is unambiguous, since if both 9' and . ...
... We will now sketch Gaifman's proof that T is complete. The LoS-Vaught test [9] says that if T has no finite models, and if every two countable models of T are isomorphic, then T is complete. Clearly, Thas no finite models. ...
Let be a finite set of (nonlogical) predicate symbols. By an -structure, we mean a relational structure appropriate for . Let be the set of all -structures with universe {1, …, n }. For each first-order -sentence σ (with equality), let μ n (σ) be the fraction of members of for which σ is true. We show that μ n (σ) always converges to 0 or 1 as n → ∞, and that the rate of convergence is geometrically fast. In fact, if T is a certain complete, consistent set of first-order -sentences introduced by H. Gaifman [6], then we show that, for each first-order -sentence σ, μ n (σ) → n 1 iff T ⊩ ω. A surprising corollary is that each finite subset of T has a finite model. Following H. Scholz [8], we define the spectrum of a sentence σ to be the set of cardinalities of finite models of σ. Another corollary is that for each first-order -sentence a, either σ or ˜σ has a cofinite spectrum (in fact, either σ or ˜σ is “nearly always“ true).
Let be a subset of which contains for each in exactly one structure isomorphic to . For each first-order -sentence σ, let ν n (σ) be the fraction of members of which a is true. By making use of an asymptotic estimate [3] of the cardinality of and by our previously mentioned results, we show that v n (σ) converges as n → ∞, and that lim n ν n (σ) = lim n μ n (σ). If contains at least one predicate symbol which is not unary, then the rate of convergence is geometrically fast.
... In 1954 Loś [27] and Vaught [53] introduced the notion of κ-categorical theory. A theory T is κ-categorical if there is only one model of T up to isomorphism. ...
We answer one of the main questions in generalized descriptive set theory, Friedman-Hyttinen-Kulikov conjecture on the Borel-reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel-reducibility notions of complexity. For any satisfying and , we show that if T is a classifiable theory and not, then the isomorphism of models of is strictly above the isomorphism of models of T with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, T, the isomorphism of models of T is either or analytically-complete.
... In particular if A and B have the same cardinality and it is greater than that of k, then they have the same dimension and so are isomorphic. So it is natural to say (as did Vaught in 1954 [214]) that a theory T is λ-categorical if it has, up to isomorphism, exactly one model of cardinality λ. Then-as Vaught noted, and we saw in section 5 that Robinson [168] had already used a version of the argument-a first-order theory which has no finite models and is λ-categorical for some λ must be complete. ...
1 The boundaries of the subject In 1954 Alfred Tarski [210] announced that 'a new branch of metamathemat-ics' had appeared under the name of the theory of models. The subject grew fast: the Omega Group bibliography of model theory in 1987 [148] ran to 617 pages. By the mid 1980s there were already too many dialects of model theory for anybody to be expert in more than a fraction. For example very few model theorists could claim to understand both the work of Zilber and Hrushovski at the edge of algebraic geometry, and the studies by Immerman and Vardi of classifications of finite structures. And neither of these lines of research had much contact with what English-speaking philosophers and European computational linguists had come to refer to as 'model-theoretic' methods or concepts. Nevertheless all these brands of model theory had common origins and important family resemblances. Some other things called models definitely lie outside the family. For example this chapter has nothing to say about 'modelling', which means constructing a formal theory to describe or ex-plain some phenomena. Likewise in cognitive science the 'mental models' of Gentner and Stephens [67] or Johnson-Laird [100] lie outside our topic. All the flavours of model theory rest on one fundamental notion, and that is the notion of a formula φ being true under an interpretation I. The classic treatment is Tarski's paper [202] from 1933. In this paper Tarski supposes that we have a language L with a precisely defined syntax. Ignoring punc-tuation, the symbols of L are of two kinds: constants and variables. The constants have fixed meanings; they will usually include logical expressions such as 'and' and 'equals'. The variables have no meaning, but (to short-circuit Tarski's very careful formulation a little) we can assign an object to each variable, and ask whether a given formula φ of L becomes true when each variable is regarded as a name of its assigned object. The grammatical categories of the variables determine what kinds of object can be assigned to them; for example we can assign individuals to individual variables, classes 1 of individuals to class variables, and so on. If A is an allowed assignment of objects to variables and A makes φ true, then A is said to satisfy φ, and to be a model of φ, and φ is said to be true in A. (In [202] Tarski says 'satisfy', and moves on to give a definition of 'true' in terms of 'satisfy'.) One of Tarski's main aims in this paper was to show that for certain kinds of language L, the relation 'A satisfies φ' is definable using only set theory, the syntax of L and the notions expressed by the constants of L. Thus one speaks of Tarski's definition of truth (or of satisfaction). In several papers around 1970, Tarski's student Richard Montague [138] set out to show that Tarski's treatment applies to some nontrivial fragments of English. This work of Montague is a paradigm of what philosophers and linguists call model-theoretic semantics. Tarski himself ([202] 6 end) foresaw this development, but he suspected that it could only be carried through by rationalising natural language to such an extent that it might not 'preserve its naturalness and [would] rather take on the characteristic features of the formalized languages'. For the rest of this chapter, we shall only be concerned with formulas of artificial languages. Tarski's 1933 paper brought into focus a number of ideas that were in circulation earlier. The notion of an assignment satisfying a formula is implicit in George Peacock ([151], 1834) and explicit in George Boole ([25] p. 3, 1847), though without a precise notion of 'formula' in either case. The word 'satisfy' in this context may be due to Edward V. Huntington (for example in [97] 1902). Geometers had spoken of gypsum or paper 'models' of geometrical axioms since the 17th century; abstract 'models' appeared during the 1920s in writings of the Hilbert school (von Neumann [147] 1925, Fraenkel [59] p. 342, 1928). In 1932 Kurt Gödel wrote to Rudolf Carnap that he was intending to publish 'eine Definition für 'wahr' ' ([73]). He never published it. We know that by 1931 Gödel already had a good understanding of definability of truth in systems of arithmetic (see Feferman [57]); there is no solid evidence on whether he had thought about general set-theoretic definitions of truth before Tarski's paper was published.
... Из 2I< 9К и Ш<Ш следует, что 21 < 91. Теорема 2.2.5* (Вот [110]). Пусть теория Т не имеет конечных моделей и категорична в некоторой бесконечной мощности у. ...
CONTENTSIntroductionChapter 1. Basic conceptsChapter 2. Decidable theoriesChapter 3. Undecidable theoriesAppendix 1. TablesAppendix 2. Some unsolved problemsReferences Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
... There is a categorical axiomatization of the real numbers with arithmetic in second order logic; this yields semantic, but not syntactic completeness of the second order theory. Vaught's proof [Vau54] of the Los-Vaught test (a first order theory with no finite models that is categorical in some infinite power is complete) writes the argument in modern terms 4 : Categoricity plus upward and downward Löwenheim-Skolem implies semantic completeness; syntactic completeness follows by Gödel. What now seem obvious compactness arguments for the existence of non-standard models were clearly not in the air in 1930 [Ken,Vau86]. ...
We approach the 'practice based philosophy of logic' by examining the practice in one specific area of logic, model theory, over the last century. From this we try to draw lessons not for the philosophy of logic but for the philosophy of mathematics. We argue in fact that the philosophical impact of the developments in mathematical logic during the last half of the twentieth century were obscured by their mathematical depth and by the intertwining with mathematics. That is, that concepts which are normally regarded by both mathematicians and philosophers as 'simply mathematics' have philosophical importance. We make two claims. First is that the mere fact that logical methods have had mathematical impact is important for any investigation of mathematical methodology. Twentieth century logic introduced techniques that were important not just for the problems they were originally designed to solve (arising out of Hilbert's program) but across broad areas of mathematics. But, from a philosophical standpoint, there is a further impact. These methods actually provide tools for the analysis of mathematical methodology.
... Because of the compactness theorem for First Order Logic this can only be the case if the model unique up to isomorphism is finite. For infinite models, J. Loś, and independently R. Vaught, [Vau57], introduced the notion of categoricity in power: A first-order theory T is categorical in an infinite cardinal κ if all of its models of cardinality κ are isomorphic. R. Vaught uses the notion to prove decidability of various theories. ...
We trace our encounters with Prof. A. Mostowski, in person and in his papers. It turns out that he and his papers left a continuous mark on my own work, from its very beginning, and till today. 1. The Z
... On the other hand, the theory of algebraically closed fields of characteristic 0 is not categorical in power No but is categorical in every higher power. In 1954 Los raised the following question: Is a theory categorical in one uncountable power necessarily categorical in every uncountable power? We give an affirmative answer to this question. ...
... Proof. The proof is based on a result of Loś and Vaught [40] which says that any first-order theory with no finite models, such that all of its countable models are isomorphic, is complete. The theory T obviously has no finite models. ...
Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences ' and `, we consider the structures with domain f1; : : : ; Ng that satisfy `, and compute the fraction of them in which ' is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kii[31] and Grove, Halpern, and Koller [22], in the general case, asymptotic conditional probabilities do not always exist, and most questions relating to this issue are highly undecidable. These results, however, all depend on the assumption that ` can use a nonunary predicate symbol. Liogon'kii [31] shows that if we condition on formulas ` involving unary predicate symbols only (but no equality or constant symbols), then the asymptotic conditional probability does exist and can be effectively computed. This is the case even if we place no corresponding restrictions on '. We extend this result here to the case where ` involves equality and constants. We show that the complexity of computing the limit depends on various factors, such as the depth of quantifier nesting, or whether the vocabulary is finite or infinite. We completely characterize the complexity of the problem in the different cases, and show related results for the associated approximation problem.
En este artıculo, realizamos una revisión de las geometrías de Zariski de dimensión uno. Comenzamos con una breve retrospectiva de conceptos clave en toria de modelos y su evolución histórica. Luego, introducimos conceptos en geometría algebraica, explorando la intersección con la teoría de modelos para motivar la teoría de modelos geométrica. Destacamos la Conjetura de tricotomía de Zilber y como la refutación de Hrushovski condujo a Zilber y Hrushovski a aislar la clase de estructuras donde la conjetura tiene validez, dando origen a las geometrías de Zariski. Concluimos presentando los resultados obtenidos por Zilber y Hrushovski en los 90s y ofrecemos una revision bibliográfica actualizada del tema.
We bring into attention the interplay between model theory of committed graphs (1-regular graphs) and their palindromic characteristic in the domain of formal languages. We prove some model theoretic properties of committed graphs and then give a characterization of them in the formal language domain using palindromes. We show in the first part of the paper that the theory of committed graphs and the theory of infinite committed graphs differ in terms of completeness. We give the observation that theories of finite and infinite committed graphs are both decidable. The former is finitely axiomatizable, whereas the latter is not. We note that every committed graph is isomorphic to the structure of integers. In the second part, as our main focus of the paper and using some of the results in the first section, we give a characterization of committed graphs based on the notion of finite and infinite palindrome strings.
We introduce the notion of limiting theories, giving examples and providing a sufficient condition under which the first order theory of a structure is the limit of the first order theories of a collection of substructures. We also give a new proof that theories like that of infinite sets are not finitely axiomatizable.
Most of the investigations contained in this essay were made for a course in logic, which I gave during the spring term of 1955 at the University of Stockholm. My intention at that time was to present a technique of logical proof that would be easier to master than those usually encountered in textbooks on logic. Thus, the essay may be regarded as having a kind of pedagogical aim. It is hoped that this aim is not overshadowed by the technical character of my exposition.
In this chapter, we intend to look at Henkin’s reviews, a total of forty-six. The books and papers reviewed deal with a large variety of subjects that range from the algebraic treatment of logical systems to issues concerning the philosophy of mathematics and, not surprisingly—given his active work in mathematical education—one on the teaching of this subject. Most of them were published in The Journal of Symbolic Logic and only one in the Bulletin of the American Mathematical Society. We will start by sorting these works into subjects and continue by providing a brief summary of each of them in order to point out those aspects that are originally from Henkin, and what we take to be mistakes. This analysis should disclose Henkin’s personal views on some of the most important results and influential books of his time; for instance, Gödel’s discovery of the consistency of the Continuum Hypothesis with the axioms of set theory or Church’s Introduction to Logic. It should also provide insight into how various outstanding results in logic and the foundations of mathematics were seen at the time. Finally, we will relate Henkin’s reviews to Henkin’s major contributions.
Model theory may be described as the union of logic and universal algebra. This chapter presents some of the methods of both western and eastern model theory. Many notions and results have two distinct versions—a western version dealing with arbitrary formulas and an eastern version dealing with quantifier-free formulas. The deeper proofs in model theory usually depend on the construction of a model with certain properties. The constructions almost always use elementary chains, diagrams and other expansions of the language, compactness theorem, downward Lowenheim–Skolem theorem, omitting type theorem, forcing, ultraproducts, and homogeneous sets. The chapter concentrates on the simplest nontrivial language, first order logic. For some applications, many-sorted logic is more natural.
This chapter presents sixteen lectures on the development of mathematical logic and the study of the foundations of mathematics in the years 1930–1964, delivered by the author in the Summer School in Vaasa, Finland in the summer of 1964. The chapter distinguishes three major movements in the philosophy of mathematics: the intuitionism of Brouwer, the logicism of Frege and Russell, and the formalism of Hilbert. The logicism of Frege and Russell tries to reduce mathematics to logic. This seemed to be an excellent program, but when it was put into effect, it turned out that there is simply no logic strong enough to encompass the whole of mathematics. Thus what remained from this program is a reduction of mathematics to set theory. The formalism of Hilbert sets up a program which requires that the whole of mathematics be axiomatized and, that these axiomatic theories be then proved consistent by using very simple combinatorial arguments. The chapter discusses the changes that the three schools underwent during the years 1930–960.
This work surveys the development of the theory of infinite ordered sets along the lines suggested by recursion theory. Since recursion theory is principally concerned with explicit effective constructions, this line of research retains much of the character of the mathematics of finite ordered sets. There are two distinct categories into which the results fall: the first is concerned with individual infinite ordered sets on which the ordering can be mechanically determined, while the second concerns whether the elementary sentences true in a given class of ordered sets can be distinguished by an algorithm. Roughly, an elementary sentence is one that refers to the elements of an ordered set rather than, say, sets of elements. “Width 3” is a notion expressible by an elementary sentence, but “finite width” cannot be expressed even by a set of such sentences. The requisite background in recursion theory and in the logic of elementary sentences can be found in Barwise [1977].
Thesis (Ph.D.)--Univ. of Minnesota. Bibliography: leaves 45-47.
Skriptum einer Vorlesung, gehalten im Wintersemester 1995/96 an der Universität Passau. Ausgearbeitet und ergänzt von Matthias Aschenbrenner.
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.
Aus: Zeitschrift f. mathematische Logik u. Grundlagen d. Mathematik. Bd 15. 1969. Maschinenschriftlich vervielfältigte Ausgabe angezeigt U 66.2728 Bonn, Math.-naturwiss. F., Diss. v. 18. Juli 1966 (Nur in beschr. Anz. f. d. Aust.).
We already defined model completeness in Chapter 1: a theory T is called model complete if every embedding between models of T is elementary. We dealt with this notion also in Chapter 2, where we considered its connection with quantifier elimination
and completeness. But now we wish to examine model completeness in a closer and more direct way, to discuss its genesis and
motivations, as well as its importance and applications.
In the literature, a type Γ is a set of formulas ϕ ( ν 0 ) in the one free variable ν 0 . A model M realizes Γ if there is an element a of M such that
M ╞ ϕ [ a ] for all ϕ ∈ Γ.
A model M omits Γ if no element of M realizes Γ. Evidently, the two statements:
(*) M realizes Γ
and
(**) M omits Γ
In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. If D is an ultrafilter over a set I , and is a structure (i.e., a model for a first order predicate logic ℒ ), the ultrapower of modulo D is denoted by D-prod . The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure (see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of . We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinal α there is an ultrafilter D over a set of power α such that for all structures , D-prod is α ⁺ -saturated.
Thesis (Ph. D.)--University of Notre Dame, February, 1971. Bibliography: l. 97-98.
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