Model-theoretic Results Research Articles

Generalizing classical and effective model theory in theories of operations and classes

Mancosu, P., Generalizing classical and effective model theory in theories of operations and classes, Annas of Pure and Applied Logic 52 (1991) 249-308. In this paper I propose a family of theories of operations and classes with the aim of developing abstract versions of model-theoretic results. The systems are closely related to those introduced and already used by Feferman for developing his program of ‘explicit mathematics’. The theories in question are two-sorted, with one kind of variable for individuals and the other for classes. The individual variables range over a domain closed under pairing and containing, among other things, natural numbers and (partial) operations. All the theories used assume a common group of axioms that insure that the individuals in the domain satisfy the conditions for a partial combinatory (applicative) algebra with pairing. We also assume a class N of natural numbers and a class induction axiom on N . Finally there are various class existence axioms. I work mainly with three theories, FMT 0, FMT and FMT Ω, which differ from each other both in their class existence axioms as well as in some further applicative axioms. These theories have various interpretations in which every object is explicitly presented by some means of definition as well as classical, or ‘standard’, set-theoretical interpretation. The systems are formulated in a flexible general language which can be used to prove abstract theorems of mathematics (analysis, algebra, model theory, etc.), thereby (depending on the system) generalizing recursive, hyperarithmetic and admissible versions of classical mathematics. The aim of my work is to give an abstract development of portions of model theory in the afore-mentioned theories, in a way to look as much as possible like classical mathematics. Thus, FMT 0 generalizes portions of countable and recursive model theory; FMT further generalizes portions of countable and hyperarithmetical model theory and finally FMT Ω provides a generalization of the classical L( Q)-completeness theorem and of an admissible version of L( Q)-completeness due to Bruce and Keisler. This continues the development of the program mentioned above, originating with Feferman.

Subformula semantics for strong negation systems

We present a semantics for strong negation systems on the basis of the subformula property of the sequent calculus. The new models, called subformula models, are constructed as a special class of canonical Kripke models for providing the way from the cut-elimination theorem to model-theoretic results. This semantics is more intuitive than the standard Kripke semantics for strong negation systems.

The non-definability notion and first order logic

The theorem to the effect that the languageLΔ introduced in [2] is mutually interpretable with the first order language is proved. This yields several model-theoretical results concerningLΔ.

Some Model‐Theoretic Results for the Relevant Logic with Quantification

Mathematical Logic QuarterlyVolume 32, Issue 19-24 p. 355-363 Article Some Model-Theoretic Results for the Relevant Logic with Quantification Mirosław Szatkowski, Mirosław Szatkowski Institute of Mathematics Pedagogical College Bydgoszcz PolandSearch for more papers by this author Mirosław Szatkowski, Mirosław Szatkowski Institute of Mathematics Pedagogical College Bydgoszcz PolandSearch for more papers by this author First published: 1986 https://doi.org/10.1002/malq.19860321909AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume32, Issue19-241986Pages 355-363 RelatedInformation

UNIVERSAL ALGEBRAIC APPROACH TO PR-DECOMPOSITION

Abstract Decomposition is one of the useful methods for investigating complex systems. In this paper, we use some universal algebraic concepts and formulate the most primary system, that is, a transition system as a universal algebra. The reason why we need the universal algebraic approach is that it makes easy to solve the preservation problems of properties of systems by using the model theoretic results. After formulating a transition system as a universal algebra (S-algebra), we discuss the PR-decomposition of S-algebras, which is a generalization of the well-known decomposition of state automata. Moreover, we solve the serial decomposition and the parallel decomposition problems of S-algebras.

Model theoretic results for infinitely deep languages

We define a subhierarchy of the infinitely deep languagesN ℵλ described by Jaakko Hintikka and Veikko Rantala. We shall show that some model theoretic results well-known in the model theory of the ordinary infinitary languages ℒℵλ can be generalized for these new languages. Among these are the downward Löwenheim-Skolem and Łoś's theorems as well as some compactness properties.

A topological analog to the Rice-Shapiro index theorem

Ever since Craig-Beth and Addison-Kleene proved their versions of the Lusin-Suslin theorem, work in model theory and recursion theory has demonstrated the value of classical descriptive set theory as a source of ideas and inspirations. During the sixties in particular, J.W. Addison refined the technique of “conjecture by analogy” and used it to generate a substantial number of results in both model theory and recursion theory (see, e.g., Addison [1], [2], [3]).During the past 15 years, techniques and results from recursion theory and model theory have played an important role in the development of descriptive set theory. (Moschovakis's book [6] is an excellent reference, particularly for the use of recursion-theoretic tools.) The use of “conjecture by analogy” as a means of transferring ideas from model theory and recursion theory to descriptive set theory has developed more slowly. Some notable recent examples of this phenomenon are in Vaught [9], where some results in invariant descriptive set theory reflecting and extending model-theoretic results are obtained and others are left as conjectures (including a version of the well-known conjecture on the number of countable models) and in Hrbacek and Simpson [4], where a notion analogous to that of Turing reducibility is used to study Borel isomorphism types. Moschovakis [6] describes in detail an effective descriptive set theory based in large part on classical recursion theory.

Concrete categories and infinitary languages

By definition, a concrete category is a category of sets which are endowed with an unspecified structure. There have been some attempts to make this structure specific. For example, Blanchard [6] used Bourbaki-type structures and Kurera and Pultr [17] have determined the structure by a functor Set-Set. Our aim is to consider concrete categories as categories of models of first-order theories. However, for these theories to be a syntactic counterpart of concrete categories, they must exceed the usual ones in the following three points: nonlogical symbols are of arbitrary arities, there may be a proper class of them and infinitary logical symbols are admitted. This language might be called ‘an unrestricted f.,,,‘. Its strength is illustrated by the fact that we may imagine any concrete category as a category consisting of models of this language. Of course, in this generality one cannot expect very model-theoretic results. Our approach is more an approach of ‘a working mathematician’, i.e. to see the relation between syntactic properties of theories and semantical properties of their categories of models. This approach has a practical aspect, i.e. to know what may be said about current models from life (e.g. when there exist limits, colimits, free objects or when models form a Cartesian closed category, a symmetric monoidal closed one etc.). There is also a theoretical aspect, i.e. to characterize theories providing a given categorical property or on the contrary to characterize concrete categories of models of theories of a prescribed kind. The pattern of these ‘categorical preservation theorems’ is the Beck-Linton theorem (see MacLane [20, p. 1471) which covers the equational case. A characteristic feature of these preservation theorems is the fact that they work outside any similarity type. The classical preservation theorem corresponding to the above mentioned Beck-Linton one is Birkhoff’s characteriz- ation of varieties of algebras of a given type. The generality of the syntax used means that the categories of models arising need not be legitimate. In Section 2 we establish a smallness property of a theory which ensures the legitimacy of its category of models. Moreover, these theories corres- pond to strongly fibre-small concrete categories in the sense of AdBmek, Herrlich 0022-4049/81/0000-OOOO/SO2.50 0 1981 North-Holland

Sheaves and Boolean valued model theory

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.

The Hanf number of the first order theory of Banach spaces

In this paper, we discuss the possibility of developing a nice i.e. first order theory for Banach spaces: the restrictions on the set of sentences for recent compactness arguments applied to Banach spaces as well as for other model-theoretic results are both natural and necessary; without them we essentially get a second order logic with quantification over countable sets. Especially, the Hanf number for sets of sentences of the first order theory of Banach spaces is exactly the Hanf number for the second order logic of binary relations (with the second order quantifiers ranging over countable sets).

Angular Distribution of Radiation Reflected from Roughened Spheres

Introduction T Note is concerned with four models for nonregular reflection from a flat plate: Lambert, Lommel-Seeliger, Euler, and the shape factor. A complete bibliography of reflectance from surfaces has been compiled by Rau and Look. This compilation includes the efforts of Russell and Schoenberg who investigated reflection from spheres using three of the aforementioned models. The fourth model is the result of an investigation whose aim was to deduce an expression that closely agreed with the experimental data of light reflected from a flat surface of pressed aluminum oxide. For the four analyses, based on the flat plate engineering intensity, equations were deduced for the radiometric intensity, the flux, and the power reflected from a roughened sphere. An experiment was then performed to obtain a limited amount of data descriptive of the radiant power bidirectionally reflected from four samples of nonregularly reflecting white spheres. The data was then compared to the four models to determine which model's theoretical results best fit the experimental data.

On certain types and models for arithmetic

AbstractThere is an analogy between concepts such as end-extension types and minimal types in the model theory of Peano arithmetic and concepts such as P-points and selective ultrafilters in the theory of ultrafilters on N. Using the notion of conservative extensions of models, we prove some theorems clarifying the relation between these pairs of analogous concepts. We also use the analogy to obtain some model-theoretic results with techniques originally used in ultrafilter theory. These results assert that every countable nonstandard model of arithmetic has a bounded minimal extension and that some types in arithmetic are not 2-isolated.

Amalgamation of polyadic algebras

The main result of the paper is that for I an infinite set, the class of polyadic I-algebras (with equality) has the strong amalgamation property; i.e., if two polyadic I-algebras have a given common subalgebra they can be embedded in another algebra in such a way that the intersection of the images of the two algebras is the given common subalgebra. Polyadic algebras were introduced by Halmos to provide an algebraic reflection of the study of first order logic without equality; later the algebras were enriched to allow the discussion of equality. That the notion is an adequate reflection of first order logic was demonstrated by Halmos' representation theorem for locally finite polyadic algebras of infinite degree (with or without equality). Daigneault and Monk have proved a strong extension of Halmos' theorem; namely, every polyadic algebra of infinite degree (without equality) is representable. Thus the notion of polyadic algebra is an adequate reflection of Keisler's predicate logic having infinitary predictates. It is an interesting question to ask for algebraic versions of various model theoretic results. Daigneault has been successful in stating and proving algebraic versions of Beth's and Craig's theorems. This was done by proving the algebraic analogue of Robinson's consistency theorem: Localy finite polyadic I-algebras (with equality) of infinite degree have the amalgamation property. The major result of the present work is to remove the locally finite condition from Daigneault's result. With the stronger result, Robinson's, Beth's, and Craig's theorems follow for Keisler's logic though we shall defer this to a later paper. We shall preface our work with an outline of the basic theory of polyadic algebras including theorems of Halmos [10] and important dilation and compression results of Daigneault and Monk [5]. Our set theoretic notation is standard, but it is perhaps worthwhile to outline some of our conventions. If X and Y are two sets, we write YX for the set of all functions from Y into X. We shall often identify 2X with Xx X-the cartesian Presented to the Society, April 8, 1967; received by the editors February 27, 1969. AMS Subject Classifications. Primary 0240, 0248; Secondary 0235, 0250.

On products of structures for generalized logics

In this paper we study structures for generalized logics and operations such as products of structures. If we consider the usual definition of products or powers of structures for the classical {0,1} ? valued logic then we see that these operations are not natural in some sense. Namely, structures which we obtain by these opera? tions have very different elementary properties. Moreover, it is also visible that even in the case of three-valued logic there is no "nice" definition of products or powers of structures. Some attempt of such construction has been given by Chang and Keis ler [2], but it can be applied only to some small class of logics. E.g. boolean-valued logics of Scott [6] do not belong to this class. The aim of this paper is to give uniform definitions of products and powers of structures for generalized logics. Our definitions seems to be natural ones in such a sense that the diagonal is an elementary substructure of a power (Theorem 4). The idea of such definition is contained implicite in the paper of Feferman and Vaught [3], see also Wojciechowska [8]. Besides, this paper contains also some model-theoretical results concerning with generalized logics. There are e.g. the Tarski?Vaught criterion for elementary in? clusion (Theorem 1) and the downward L?wenheim?Skolem Theorem (The? orem 2).

On the model theory of denumerably long formulas with finite strings of quantifiers

In this paper we prove infinitary analogues of model-theoretic results known for finitary logic. The infinitary language we deal with is Lω1ω which is roughly described by saying that, in addition to the usual formation rules of the lower predicate calculus with identity, also the formation of the conjunction and disjunction of countably many formulas is allowed.

Some model theoretic results for θ-logic

The θ-completeness theorem is applied to prove theorems above two-cardinal models, homogeneous models, and categoricity in power in θ-logic.

Rings of continuous integer-valued functions and nonstandard arithmetic

0. Introduction. In this paper rings of continuous integer-valued functions are studied, with particular attention paid to their maximal residue class domains. These domains correspond bijectively to minimal prime ideals, rendering the space of these ideals of particular interest. Since these domains are either the integers or are nonstandard models of the integers, questions about nonstandard arithmetic will also be considered. In ?1 the space of minimal prime ideals of C(X, Z), the ring of continuous functions from a nonempty Hausdorff space X into Z, the ring of integers, is showed to be homeomorphic to 6X (1.2), the Boolean space of the algebra of open-and-closed sets of X. The maximal ideal space of C(X, Z) is shown to map continuously onto 6X (1.3). The space, 6OX, of points of 3X that give rise to integer residue class domains, is studied in ?2. The map of X into boX strongly resembles the realcompactification injection [GJ]. A representation theorem of C(X, Z) over 3OX is also given (2.4). It is shown in ?3 that points in 3X 6OX give rise to Z, a nonstandard model of Z (3.1). Here some of the relevant background material in model theory is discussed. The algebraic theory of nonstandard arithmetic is studied in ?4. In ?5 we return to study Z, its maximal ideal space, and its quotient field Q, which is a nonstandard model of the rational field Q. In ?6, the most technical section of the paper, the valuations of Q associated with maximal ideals of Z are computed (6.3). The value groups that arise are analysed ((6.4), (6.5), and (6.6)), followed by some rather striking results in case the maximal ideal in question is principal. The ideals of Z are analyzed in ?7 along classical lines: i.e., we proceed from the study of maximal and prime ideals, through the study of primary ideals, to a decomposition theorem for ideals in terms of primary ideals (7.4). Ideals in C(X, Z) are decomposed in ?8, first into coprimary ideals (8.4), and then into primary ideals (8.9). In the process, the sets of maximal, prime, coprimary, and primary ideals of C(X, Z) are analyzed. In ?9 some model-theoretic results are obtained on the residue class fields of C(X, Z), the principal result being that any such field is elementarily equivalent

On automorphisms of polyadic algebras

Introduction. This paper belongs to the theory of polyadic algebras as developed by Halmos [11-15], but it has a bearing on the theory of models. Two central concepts are those of homogeneous(2) and of normal extensions of a polyadic algebra (see the beginning of ?2 and of ?6). These concepts bear some resemblance to concepts of the theory of algebraic extensions of fields; however, we have been influenced more immediately by M. Krasner's general Galois theory [33-35] which can be given a polyadic interpretation. Aside from the short preliminary Chapter 0, the paper is divided into sections grouped into three chapters. Each section begins by an outline of its contents. We proceed to an analysis of the main results of the paper. All polyadic algebras considered are locally finite of infinite degree. The highlights of Chapter I are: (i) the possibility of extending any simple extension of a (simple polyadic) algebra to a simple homogeneous and normal extension (Corollary 4.6); (ii) a first step on a Galois theory (Theorem 4.5) which, in the case of a full simple functional algebra with finite domain, takes a definite form (Theorem 4.7) closely related to Krasner's theory in that case; (iii) the existence and unicity of o+-universal-homogeneous algebras in the sense of B. Jonsson [21; 22] in certain classes of polyadic algebras (Theorem 6.3, Theorem 6.6). The results in (iii) make use of Jonsson's work which is dependent on the continuum hypothesis (see also Theorem 5.5 which is independent of this hypothesis). Chapter If departs from the theme of automorphisms and uses little aside from Chapter 0 and ?1. The main result here is Theorem 8.1 which gives a description of the functional representations of a full simple functional algebra in terms of a new generalization of the concept of reduced power. In Chapter III, we propose to show that the algebraic results of the first two chapters, when put together, embody several known model-theoretic results or new forms of such results. After a section devoted to the connections between Model theory and polyadic algebras, we give new proofs of Beth's theorem