Mathematik-Klassifikation / Teil 2 AMS 03Cxx model theory AMS 03C* (including sublevels) AMS 03C55 Set-theoretic model theory; AMS 03C57 Recursion-theoretic model theory http://www.ub.uni-heidelberg.de/helios/fachinfo/www/math/ams2.htm
Extractions: AMS: 01-XX Not classified at a more specific level AMS: 01-00 General reference works AMS: 01-01 Instructional expositions AMS: 01-02 Research expositions AMS: 01-06 Proceedings, conferences, etc. AMS: 01A05 General histories, source books AMS: 01A15 Indigenous cultures of Europe, pre-Greek AMS: 01A20 Greek, Roman AMS: 01A25 China AMS: 01A30 Islam (Medieval) AMS: 01A35 Medieval AMS: 01A40 15th and 16th centuries, Renaissance AMS: 01A45 17th century AMS: 01A50 18th century AMS: 01A55 19th century AMS: 01A60 20th century AMS: 01A70 Biographies, obituaries, personalia, bibliographies AMS: 01A72 Schools of mathematics AMS: 01A73 Universities AMS: 01A75 Collected or selected works; reprinting or translations of classics
Extractions: In this introductory article, we first look at some motivational issues, such as the need for plug-in compatible components and the different ways in which compatibility can be judged. Reasons for studying object-oriented type theory include the desire to explain the different features of object-oriented languages in a consistent way. This leads into a discussion of what we really mean by a type, ranging from the concrete to the abstract views. The eventual economic success of the object-oriented and component-based software industry will depend on the ability to mix and match parts selected from different suppliers [1]. In this, the notion of component compatibility is a paramount concern: the client (component user) has to make certain assumptions about the way a component behaves, in order to use it;
Extractions: Home Wesley Phoa Abstract: A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical `universe' of recursive mathematics. Among its objects are the modest sets , which form a set-theoretic model for polymorphism. More precisely, there is a fibration of modest sets which satisfies suitable categorical completeness properties, that make it a model for various polymorphic type theories. These lecture notes provide a reasonably thorough introduction to this body of material, aimed at theoretical computer scientists rather than topos theorists. Chapter 2 is an outline of the theory of fibrations, and sketches how they can be used to model various typed lambda-calculi. Chapter 3 is an exposition of some basic topos theory, and explains why a topos can be regarded as a model of set theory. Chapter 4 discusses the classical PER model for polymorphism, and shows how it `lives inside' a particular topos - the effective topos - as the category of modest sets. An appendix contains a full presentation of the internal language of a topos, and a map of the effective topos. Chapters 2 and 3 provide a sampler of categorical type theory and categorical logic, and should be of more general interest than Chapter 4. They can be read more or less independently of each other; a connection is made at the end of Chapter 3.
1 Introduction Tarskian model theory is almost universally understood as a formal is to furnish its Settheoretic interpretation in a suitable model structure; http://www.hf.uio.no/ifikk/filosofi/njpl/vol2no1/models/node1.html
Extractions: Next: 2 Language and the Up: Language and its Models:Is Previous: Language and its Models:Is Tarskian model theory is almost universally understood as a formal counterpart of the preformal notion of semantics, of the ``linkage between words and things''. The wide-spread opinion is that to account for the semantics of natural language is to furnish its set-theoretic interpretation in a suitable model structure; as exemplified by Montague 1974 The thesis advocated in this paper is that model theory cannot be considered as semantics in this straightforward sense. We try to show that model theory is more adequately understood as shining light on considerations concerning the relation of consequence than on those concerning the relation of expressions to extralinguistic objects; and that it makes little sense to use model theory for the purposes of answering such questions as what is meaning? or when is a sentence true?. The organization of the paper is the following: We start by considering various formal reconstructions of natural languages utilizing standard logic. Section points out that the usual way of explicating the semantics of natural language, namely the way of Tarskian model-theoretic interpretation, is problematic. In Section
Vita Logic and algebra (model theory; module theory; applications of model theory Lecture series, NSFsupported workshop on Set-theoretic methods in algebra http://www.math.uci.edu/faculty/peklofv.html
Extractions: Vita Vita in HTML format Paul C. Eklof EDUCATION A.B., Columbia College New York New York Ph.D., Cornell University Ithaca New York RESEARCH AREA Logic and algebra (Model theory; module theory; applications of model theory and set theory to algebra) PROFESSIONAL EXPERIENCE Gibbs Instructor in Mathematics, Yale University , New Haven, Connecticut, 1968-1970 Assistant Professor of Mathematics, Stanford University , Stanford, California, 1970-1973 Associate Professor of Mathematics, University of California, Irvine, 1973-1978 Visiting Associate Professor of Mathematics, Yale University , New Haven, Connecticut, 1975-1976 Visiting Professor of Mathematics, Simon Fraser University Burnaby , B.C., Jan.- July 1985 Professor of Mathematics, University of California Irvine Professor Emeritus of Mathematics, University of California Irvine INVITED LECTURES
Barry Jay's Research Interests: Shape Theory My main area of research is in Shape theory and its applications in programming and yet we know that system F has no Settheoretic model, at least, http://www-staff.it.uts.edu.au/~cbj/Publications/shapes.html
Extractions: Loops Order Enriched Categories Internal Languages My main area of research is in Shape Theory and its applications in programming language design. This work is listed under the following sub-headings (brackets enclose related language designs). Shape in Computing is a position paper on the importance of shape in computing. It begins: "Values associated with a data type usually have a shape, too. For example, the shape of a matrix is its size. Shape refers to data structures into which data can be inserted at various positions, or ``holes''. For example, the shape of a labelled graph is its underlying, unlabelled graph; that of a record is its set of field names. Shapes, like types, are important in the specification and semantics of programs, and should support tools for static error detection and improved compilation techniques. This position paper will outline the semantic underpinnings of shape theory, and suggest how it might be exploited in a variety of computational settings." Separating Shape from Data is the summary of an invited address emphasising the semantic aspects of shape. It begins:
Re: SUO: Composing Ontologies Using Morphisms And Colimits The punch line is that CATEGORICAL model theory IS DISTINCTLY DIFFERENT FROM Settheoretic model theory, AND THAT IS WHY I Mike Healy SAY THAT THE http://grouper.ieee.org/groups/suo/email/msg02821.html
Extractions: Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index To sowa@bestweb.net Subject : Re: SUO: Composing Ontologies using morphisms and colimits From mfu@redwood.rt.cs.boeing.com Date : Tue, 16 Jan 2001 18:39:23 -0800 (PST) CC mjhealy@puffin.rt.cs.boeing.com Matthew.R.West@is.shell.com phayes@ai.uwf.edu rekent@ontologos.org ... standard-upper-ontology@ieee.org Reply-To mfu@redwood.rt.cs.boeing.com Sender owner-standard-upper-ontology@ieee.org Prev by Date: SUO: RE: Re: Revised definition of Base Document Next by Date: SUO: RE: RE: Vote on Merged Ontology as a 'Base Document' for SUO WG Prev by thread: Re: SUO: Composing Ontologies using morphisms and colimits Next by thread: Re: SUO: Composing Ontologies using morphisms and colimits Index(es): Date Thread
Extractions: This Article Extract FREE Full Text (PDF) Alert me when this article is cited ... Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive ... Request Permissions Google Scholar Articles by Jané, I. Search for Related Content Philosophia Mathematica Ignacio Jané Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona 08028 Barcelona, Spain When we encounter a theorem with a composite name, like Heine-Borel, Cantor-Bendixson, or Löwenheim-Skolem, we are curious to know what the particular contribution to it of each author actually was. The obvious guess is an alternative: either the first author
The Homepage Of The Helsinki Logic Group Taneli Huuskonen , docent, model theory, set theory, logic and analysis set theoretic model theory, e.g. transfer principles and universality of regular http://www.logic.math.helsinki.fi/
Extractions: Logic Colloquium 2003: Group photo and lecture materials Members Research Publications ... Contact Info Members - Research Publications Links Contact Info ... Aapo Halko , Ph.D., descriptive set theory Alex Hellsten , Ph.D., set theory Taneli Huuskonen , docent, model theory, set theory, logic and analysis Tapani Hyttinen , docent, stability theory, infinitary logic Juliette Kennedy , docent, models of arithmetic, philosophy of mathematics Meeri Kesälä , Ph.D., model theory Juha Kontinen , Ph.D., finite model theory Kerkko Luosto , docent, finite and infinite model theory, abstract model theory Juha Oikkonen , university lecturer, infinitary logic, nonstandard analysis Matti Pauna , Ph.D. Juha Ruokolainen , Ph.D. , professor, finite model theory, abstract model theory, set theory Tapio Eerola , M.Sc. , Ph.L. Jarmo Kontinen , M.Sc. Hannu Niemistö , Ph.L., finite model theory Ville Nurmi , M.Sc. Ryan Siders , M.Sc. Former members of the group can be found in the list of Ph.Ds
[FOM] Explicit Construction; Choice And Model Theory It is not responsive to the basic point that the set theoretic structure of the domain is just NOT model theory. I think that Angus and Lou would agree with http://cs.nyu.edu/pipermail/fom/2003-July/006940.html
Extractions: Wed Jul 2 20:32:30 EDT 2003 Reply to Baldwin 12:11PM 7/1/03. This interchange is really about what model theory might look like if one pays systematic attention to certain foundational matters surrounding explicitness. My thinking is that this point of view is not at odds with the current trends I see in papers and meetings in current model theory. NOTE: This foundational interchange should not slow down any plans for your presentations of model theoretic material. Let T be a first order sentence that has an infinite model. I have been interested in the question of whether you can explicitly construct a model of T whose domain is a given infinite set D. Now I understand what seems to me a very strange question. Here is the obvious response.
Axiomatic Set Theory - Wikipedia, The Free Encyclopedia In this approach it is demonstrated that a particular statement in set theory can be used to prove the existence of a set model of ZFC and thereby http://en.wikipedia.org/wiki/Axiomatic_set_theory
Extractions: var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Jump to: navigation search In mathematics axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory . The basis of set theory was created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (numbers, functions, etc.,) from all the traditional areas of mathematics ( algebra analysis topology , etc.) in a single theory, and provides a standard set of axioms to prove or disprove them. At the same time the basic concepts of set theory are used throughout mathematics, the subject is pursued in its own right as a specialty by a comparatively small group of mathematicians and logicians . It should be mentioned that there are also mathematicians using and promoting different approaches to the foundations of mathematics. The basic concepts of set theory are set and membership. A
Basic Model Theory, Kees Doets Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.springerlink.com/index/V10576212J886801.pdf
Re: [ontolog-forum] Ontological Correctness Just as we can define the numbers to be set theoretic objects, If the models are axiomized under ZF, then the model theory is just a mapping from http://ontolog.cim3.net/forum/ontolog-forum/2007-02/msg00022.html
Extractions: ontolog-forum Top All Lists Date Advanced ... Thread from [ Chris Menzel Permanent Link Original To From cmenzel@xxxxxxxx Date Fri, 2 Feb 2007 15:51:54 -0600 Message-id 20070202215154.GA589@xxxxxxxx Chris, If the models are axiomized under ZF, then the model theory is just a mapping from one set of axioms to another, which of course can be axiomitized. Conrad, I don't understand this response. If the model theory of a language is axiomatized under ZF (I'm not sure it means to axiomatize a *model*), the model theory is NOT a mapping from one set of axioms to another. It is a theory, expressed within ZF, of the formal relations between languages and their interpretations and consequently between theories and their models. correctness, unless you happen to feel comforable with the axiomitization of the model. By "axiomatization of the model" I take you to mean something like "definition of the class of models" of a given theory. I guess I'm not sure what the point is here. It is possible of course that one might define a class of intended models and then, lacking a formal proof of the fact, doubt whether one's theory picks out exactly that class. Sure, that might happen. But so what? There are plenty of cases where we're quite confident of the model theory and hence where various notions of correctness might have some purchase, e.g., PSL. Indeed, if you *can't* describe the intended models of a theory mathematically, it seems to follow that you might not be terribly sure that you know what your theory is describing in the first place. (This was in fact a real problem for the untyped lambda calculus until the great Dana Scott came up with domain theory.)
ScienceStorm - Inner Models, Fine Structure And Large Cardinals The last area focuses on the theory of inner models; the main objective here is to Inner model theory, forcing, descriptive set theory, and infinitary http://www.sciencestorm.com/award/0500799.html
Extractions: National Science Foundation Award #0500799 Investigator(s): Martin Zeman (PI) Sponsor: University of California-Irvine , CA 92697 9498244768 Start Date/Expiration Date 2005-07-01 to 2006-06-30 (amended 2005-04-06) Awarded Amount to Date: Abstract: NSF Org: DMS - Division of Mathematical Sciences Award Number: Award Instrument: Continuing grant Program Manager: Tomek Bartoszynski
Translinguistic Poetica: Model Theory Contributions model theory contributions. Let L be a language. circumstance of the subject, C, contain inference rules or the syntactical or set theoretic framework http://transpoetics.blogspot.com/2007/10/model-theory-contributions.html
Extractions: @import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?targetBlogID=34549658"); THE POET IS A SCIENTIST DO NOT UNDERSTAND. BECOME UNDERSTAND. Let L be a language. Let L contain abstractions, logical and non logical symbols, and a grammar wherein those syntactic components in logical and non logical symbols are a metalanguage or a sort of informal set theory and, as will later be noted, provide a secondary metalanguage, that is to be later discussed, and labeled with (e). Let (A), an abstraction be any [category] of [subjects] or that dedition of all the indices of partitions of free variables expected by those subjects, such that an abstraction may suggest a particular taxonomy amongst those partitions of free variables as a subject. Let a subject be that nomination of certain partitions or free variables A and non logical operators that will be called a [circumstance] of those partitions or free variables in the L-structure. Let a subject be a free variable on an L-structure as an encoding of taxonomy of some other free variables obtained by the instrumentality of the category of subjects in A so that we let a subject be which some of who's partitions are found in Asymptotic knowledge, and let all the partitions of free variables in a subject be Symplectic knowledge.
5.4.1 Type (a Further Discussion) using standard model theory, in terms of the set of all interpretations Our goal is to define some set theoretic entity that captures this intuition. http://www.joaquin.net/cuml/2torial/type-formal.htm
Extractions: - for type checking. A model identifies a type by providing a name for the type, which is a one place predicate name. Some types are left undefined in the model, used by the modeler as primitives, while other types are defined. The definition of a type takes the form of a statement with two parts. Each part is a statement. The two parts are joined by the connective, ' means The first part, the defined expression , contains the constant predicate name for the type, and applies that to exactly one variable name; the second part, the type specification , is an open statement with exactly the same variable free (with that variable free, and no other variable free). Thus, the syntax of a type definition is:
Pref Thus certain things which were done in model theory and its nonstandard theories which incorporate more of the set theoretic instrumentarium one is http://www.math.uni-wuppertal.de/~reeken/pref.html
Extractions: In the aftermath of the discoveries in foundations of mathematics there was surprisingly little effect on mathematics as a whole. If one looks at standard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice ( ZFC ) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathematical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at ``unnatural'' interpretations of the membership relation. One of the crucial discoveries in foundations was that the structures studied in mathematics do have nonstandard models. Starting with A.Robinson, this gave rise to a new mathematical discipline
Set Theory & The Euclidean Model Set theory Euclidean model for the codification of mathematics The set theoretic foundation relied on the thoughtbased methods of logic, that is, http://whyslopes.com/volume1a/ch14a_Set_Theory_Mathematics.html
Set Theory - Search Radar model theory This volume is an introduction to inner model theory, inner models reflecting large cardinal properties of the set theoretic universe. http://searchradar.webaroo.com/s?searchQuery=Set Theory
