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Topic: Model theory

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Set theory

Axiomatic set theory

Saturated model

Elementary embedding

Proof theory

Compactness theorem

Mathematical model

Default logic

Kripke semantics

Axiomatic system

Set

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	Encyclopedia: Model theory (Site not responding. Last check: 2007-11-07)
		The independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory (proved by Paul Cohen and Kurt Gödel) are the two most famous results arising from model theory.
		In the past decade, model theory has reached a new maturity, strengthening of its connections with other areas of mathematics and producing striking applications to diophantine geometry, analytic geometry and Lie theory, as well as strong interactions with group theory, representation theory of finite-dimensional algebras, and the study of the $p$-adics.
		The semester-long program on the model theory of fields at MSRI in 1998 brought together model theorists, number theorists and algebraic and analytic geometers, and set a new precedent for communication between the various groups.
	www.nationmaster.com /encyclopedia/Model-theory (906 words)

	Introduction
		Model theory is the study of the interpretations of any language, formal or natural, by means of set-theoretic or category-theoretic structures.
		First-order model theory is the branch of mathematics that deals with the relationships between descriptions in first-order languages and the structures that satisfy these descriptions.
		In support of the IFF Model Theory Ontology are the two lower levels of this structure, which are represented by the IFF Lower Classification Ontology (middle level) and the IFF Lower Core Ontology (lower level).
	suo.ieee.org /IFF/versions/20020515/IFFModelTheoryOntology/Introduction.htm (373 words)

	Model Theory
		Model theory began with the study of formal languages and their interpretations, and of the kinds of classification that a particular formal language can make.
		But in a broader sense, model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Alfred Tarski's truth definition as a paradigm.
		These are very different senses of ‘model’ from that in model theory: the ‘model’ of the phenomenon or the system is not a structure but a theory, often in a formal language.
	plato.stanford.edu /entries/model-theory (6225 words)

	Learn more about Model theory in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		A model is formally defined in context of some language L, following Tarski's concept of truth.
		Model theory is concerned with first order logic, and to first order logic all cardinals look the same.
		This is expressed in the Lowenheim-Skolem theorems - which state that any theory with an infinite model A has models of all infinite cardinalities (at least that of the language) which agree with A on all sentences - they are "elementarily equivalent".
	www.onlineencyclopedia.org /m/mo/model_theory.html (797 words)

	First-order Model Theory
		First-order model theory, also known as classical model theory, is a branch of mathematics that deals with the relationships between descriptions in first-order languages and the structures that satisfy these descriptions.
		From another point of view, first-order model theory is the paradigm for the rest of model theory; it is the area in which many of the broader ideas of model theory were first worked out.
		The programme is broadly to classify structures according to (a) what groups or fields are interpretable in them (in the sense sketched in the entry on model theory) and (b) whether or not the structures have ‘modular geometries’; and then to use this classification to solve problems in model theory and geometry.
	plato.stanford.edu /entries/modeltheory-fo (6179 words)

	RDF Model Theory
		The model theory assigns interpretations directly to the graph; we will refer to this as the 'graph syntax' to avoid ambiguity, since the bare term 'syntax' is often assumed to refer to a lexicalization.
		However, the semantic model given here distinguishes properties and classes as objects from their extensions - the sets of object-value pairs which satisfy the property, or things that are 'in' the class - thereby allowing the extension of a property or class to contain the property or class itself without violating the axiom of foundation.
		In this version of the model theory, literals are treated as simple names which refer to literal values, and no particular assumptions are made about the role of datatype specifications.
	lists.w3.org /Archives/Public/www-archive/2002Jan/att-0007/01-RDF_Model_Theory.htm (7128 words)

	RDF Semantics
		model theory for specifying the semantics of a formal language.
		Readers unfamiliar with model theory may find the glossary in appendix B helpful; throughout the text, uses of terms in a technical sense are linked to their glossary definitions.
		An alternative way to specify a semantics is to give a translation from RDF into a formal logic with a model theory already attached, as it were.
	www.w3.org /TR/rdf-mt (11716 words)

	Semantic Web Model Theory (Site not responding. Last check: 2007-11-07)
		Model theory is usually most relevant to implementation via the notion of entailment, described later, and by making it possible to define valid inference rules.
		One approach is for each of the SWELs to be defined in terms of their own model theory, layering it on top of the model theories of the languages they are layered upon.
		The important point to note about the avove diagram is that if the Li to Lbase mapping and model theory for Li are done consistently, the Li model theory interpretations satisfying G will be the same as the Lbase model theory interpretations satisfying G (modulo the richer interpretations not in Li's MT).
	tap.stanford.edu /SemanticWebSemantics.html (3132 words)

	The Five-Factor Model
		The five-factor theory is among the newest models developed for the description of personality, and this model shows promise to be among the most practical and applicable models available in the field of personality psychology (Digman, 1990).
		Although the five-factor model leaves much to be desired as far as the explanation of the numbers, it was shown that with the sliding scales associated with each of the five variables, the five-factor model was easily quantifiable.
		As this model does not quite live up to the standards for a great theory in personality (it would be tough to find a theory that does) perhaps a more appropriate name for it would be a great taxonomy in personality.
	www.personalityresearch.org /papers/popkins.html (6128 words)

	M-theory, the theory formerly known as Strings
		The standard model was designed within a framework known as Quantum Field Theory (QFT), which gives us the tools to build theories consistent both with quantum mechanics and the special theory of relativity.
		Apart from the fact that instead of one there are five different, healthy theories of strings (three superstrings and two heterotic strings) there was another difficulty in studying these theories: we did not have tools to explore the theory over all possible values of the parameters in the theory.
		Each theory was like a large planet of which we only knew a small island somewhere on the planet.
	www.damtp.cam.ac.uk /user/gr/public/qg_ss.html (1344 words)

	Complexity Theory and Finite Model Theory
		Computational complexity theory is concerned with the investigation of the resources (usually time and space) required for the computational solution of problems.
		Finite model theory is the model theory of finite structures, where model theory (of finite and infinite structures) is the branch of mathematical logic dealing with the relationship between a formal language and its interpretation in mathematical structures: it has a long and varied history.
		The model theory of finite structures is very underdeveloped although interest in the subject has now exploded, mainly due to the intimate relationship between finite model theory and complexity theory.
	www.swan.ac.uk /compsci/ResearchGroups/TheoryGroups/CTFMT.html (178 words)

	RDF Model Theory (Site not responding. Last check: 2007-11-07)
		precise semantic theory for RDF and RDFS, and to sharpen the notions of consequence and inference.
		To apply the model theory to this kind of situation, one should think of the assertion of such a graph as amounting to an assertion of the merge of that graph together with whatever RDF graphs are assumed to define the public vocabulary, in order to fully convey the intended meaning of the 'public' assertion.
		So far we have considered only the model theory of what might be called the logical form of the RDF graph itself, without imposing any special interpretations on any reserved vocabulary.
	www.w3.org /TR/2002/WD-rdf-mt-20020429 (11311 words)

	William Weiss and Cherie D'Mello, Fundamentals of Model Theory
		Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics.
		This book provides an introduction to Model Theory which can be used as a text for a reading course or a summer project at the senior undergraduate or graduate level.
		More importantly, the methods of Model Theory display clearly the structure of the main ideas of the proofs, showing how theorems of logic combine with theorems from other areas of mathematics to produce stunning results.
	at.yorku.ca /i/a/a/i/10.htm (647 words)

	CATASTROPHE THEORY MODEL OF THE CONFLICT HELIX
		This is a fundamental assumption in the helix and for the catastrophe model.
		It should be clear that Figure 5 is a (to borrow the label often applied to topology) rubber sheet model, and that the actual position and rotation of a cusp for the history of a specific dyad may differ considerably from the picture.
		This is because the random sample (from a population) model is inappropriate, as is the combinatorial model (where the results are conceived of as one possible random combination among all the possible combinations of the given data).
	www.hawaii.edu /powerkills/CAT.ART.HTM (8805 words)

	RDF Model Theory
		The model theory assigns interpretations directly to the graph, which is taken as being the primary RDF syntax, in the sense that two RDF documents, in whatever lexical form, are syntactically equivalent if and only if they map to the same RDF graph.
		As stated here, the model theory only supports interpretations in which containers are 'opaque' objects, so that assertions involving containers are about those containers, rather than being understood 'transparently' to be asserting anything about the members of the containers.
		With the current model theory, this says that the value of foo is baz for some thing in the Alt container, which might be aaa, bbb or something else not mentioned in the graph.
	blogspace.com /rdf/modeltheory (4775 words)

	Theory: Model or Theory?
		Scientist do not use the term "the theory of.." except for those ideas that have been so thoroughly tested and developed that we know there is indeed some range of phenomena for which they give correct predictions every time.
		Such a theory can never be proved to be complete and final -- that is why we no longer call it a "law." However, it is the same kind of well-tested set of rules, with an established area of applicability, as the older ideas called "laws".
		Similarly, the theory of atoms is not invalidated, but rather extended, by discoveries of structure within protons and neutrons.
	www2.slac.stanford.edu /vvc/theory/modeltheory.html (555 words)

	Realism, model theory, and linguistic semantics (Site not responding. Last check: 2007-11-07)
		On this approach the semantics for a language must first define what a model or interpretation for the language is -- minimally a domain of individuals and a function assigning referents from that domain to the language's nonlogical constants (essentially the content words, in the case of a natural language).
		For suppose we have an epistemically perfect or ideal theory of the world, T. T is consistent with all the observations we will ever be able to make and satisfies all of our theoretical constraints.
		It is such an obvious requirement, that it is usually taken for granted in empirical semantic theories and not stated explicitly.
	members.aol.com /lshauser/mts.html (3289 words)

	INI Programme MAA Satellite Workshop - Pure Model Theory (Site not responding. Last check: 2007-11-07)
		There is a rich interplay between the sophisticated and difficult techniques which have been developed internally within model theory (so-called 'pure model theory') and the model-theoretic analysis of individual (or groups of) mathematical structures which arise in algebra and analysis.
		This workshop will focus on developments at the 'purer' end of model theory, without, of course, ignoring the connections with other branches of mathematics.
		The themes of the workshop will be: forking and simple theories; Hrushovski constructions; non-elementary classes; topological methods in model theory; independence in unstable structures.
	www.newton.cam.ac.uk /programmes/MAA/maaw02.html (286 words)

	Theory of Reasoned Action/Planned Behavior Overview (Site not responding. Last check: 2007-11-07)
		In 1988, the Theory of Planned Behavior (TPB) was added to the existing model of reasoned action to address the inadequacies that Ajzen and Fishbein had identified through their research using the TRA.
		This theory is then applied to the assessment of how this affects the attitudes and behaviors of persons with regards to grassroots organizations.
		The Theory of Planned Behavior is applied to the study of how financial contribution or commission to sales agents affect their attitudes towards a certain behavior.
	hsc.usf.edu /~kmbrown/TRA_TPB.htm (3144 words)

	Shannon-Weaver model
		The model was produced in 1949, a year after Lasswell's and you will immediately see the similarity to the Lasswell Formula.
		In the above discussion of the model I have often referred to meaning, a topic largely absent from the original model, but it is only by broadening the model to take in meaning and the biological, cognitive, technological, socio-cultural and other factors which influence it that this model can be of any use.
		Perhaps one of the main reasons for the model's popularity amongst communication theorists in the 'humanities' has been that it provides them with a ready-made jargon that ordinary mortals are not likely to be familiar with, as well as conferring on the subject a kind of pseudo-scientific respectability.
	www.cultsock.ndirect.co.uk /MUHome/cshtml/introductory/sw.html (3866 words)

	Model Theory. Goedel's Completeness Theorem. Skolem's Paradox. Ramsey's Theorem. By K.Podnieks
		Model theory is investigation of formal theories in the metatheory ZFC.
		They thought that mere consistency of a theory (in the syntactic sense of the word - as the lack of contradictions) is not sufficient to regard a theory as a "meaningful" one.
		Model Existence theorem is steadily provoking the so-called Skolem's paradox.
	www.ltn.lv /~podnieks/gta.html (5980 words)

	INI Programme MAA Workshop - Model Theory, Algebraic and Analytic Geometry
		The main themes are o-minimality (from model theoretic, real analytic and computational viewpoints), Diophantine geometry (including Hilbert's 10th problem),rigid analytic geometry and motivic integration.
		The currently invited speakers are as listed below but as this workshop coincides with the last two weeks of the Model Theory Semester, we have left several slots available for the presentation of results obtained at the Institute during the course of the previous six months.
		Model Theory and Applications to Algebra and Analysis
	www.newton.cam.ac.uk /programmes/MAA/maaw03.html (322 words)

	Model Theory of Fields: Suggested Reading (Site not responding. Last check: 2007-11-07)
		A survey on the model theory of the real numbers with exponentiation that appeared in the Notices of the AMS.
		D. Marker, An introduction to the Model Theory of Fields.
		An introduction to geometric model theory and Hrushovski's proof of the Mordell-Lang conjecture.
	www.math.uic.edu /~marker/mtf-reading.html (125 words)

	SSRN-The Capital Asset Pricing Model: Theory and Evidence by Eugene Fama, Kenneth French
		Before their breakthrough, there were no asset pricing models built from first principles about the nature of tastes and investment opportunities and with clear testable predictions about risk and return.
		Unfortunately, perhaps because of its simplicity, the empirical record of the model is poor - poor enough to invalidate the way it is used in applications.
		We then review the history of empirical work on the model and what it says about shortcomings of the CAPM that pose challenges to be explained by more complicated models.
	papers.ssrn.com /sol3/papers.cfm?abstract_id=440920 (507 words)

	"Midwest Model Theory"
		Abstract: Scott rank is a measure of model theoretic complexity.
		These can be used to separate between continuous theories which produce counter-examples to many classical results and theories for which we have hope of extending the classical theory.
		As with the $\infty$-definable group $p^\infty A (K)$ of Hrushovski's proof, the arithmetic and model theory of a certain $\infty$-definable group, $\phi^\sharp$, controls the arithmetic of a Drinfeld module $\phi$.
	www.math.uiuc.edu /ResearchAreas/logic/conference/MWMT (923 words)

	OUP: Model Theory: Manzano (Site not responding. Last check: 2007-11-07)
		Logic languages are free from the ambiguities of natural languages, and are therefore specially suited for use in computing.
		Model theory is the branch of mathematical logic which concerns the relationship between mathematical structures and logic languages, and has become increasingly important in areas such as computing, philosophy and linguistics.
		As the reasoning process takes place at a very abstract level, model theory applies to a wide variety of structures.
	www.oup.co.uk /isbn/0-19-853851-0 (320 words)

	Model Theory. Skolem's Paradox. Ramsey's Theorem.
		Model Existence theorem says that (syntactic) consistency of a theory is sufficient to regard it as "meaningful": if a theory does not contain contradictions, then it describes at least some kind of "mathematical reality".
		Idea #3: prove the (non-constructive) Lindenbaum's lemma: any consistent theory has a consistent complete extension (the axiom set of the extension may not be effectively solvable).
		Some 30 years after Goedel's proof, in 1963 P.Cohen proved that if ZFC is a consistent theory, then this theory is not able to solve Cantor's continuum problem.
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gta.html (5899 words)

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