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Model theory

Model theory
This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. Universal algebra[edit] and or . . Related:  The problems with philosophy

Higher-order logic Formal system of logic The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types (see Type theory). Leon Chwistek and Frank P. Quantification scope[edit] First-order logic quantifies only variables that range over individuals; second-order logic, in addition, also quantifies over sets; third-order logic also quantifies over sets of sets, and so on. Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply. Semantics[edit] There are two possible semantics for higher-order logic. In the standard or full semantics, quantifiers over higher-type objects range over all possible objects of that type. In Henkin semantics, a separate domain is included in each interpretation for each higher-order type.

Computability theory Study of computable functions and Turing degrees Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages. Introduction[edit] Turing computability[edit] The terminology for computable functions and sets is not completely standardized. Areas of research[edit]

Structure Arrangement of interrelated elements in an object/system, or the object/system itself Load-bearing[edit] Buildings, aircraft, skeletons, anthills, beaver dams, bridges and salt domes are all examples of load-bearing structures. The results of construction are divided into buildings and non-building structures, and make up the infrastructure of a human society. The structure elements are combined in structural systems. Load-bearing biological structures such as bones, teeth, shells, and tendons derive their strength from a multilevel hierarchy of structures employing biominerals and proteins, at the bottom of which are collagen fibrils.[4] Biological[edit] In another context, structure can also observed in macromolecules, particularly proteins and nucleic acids.[6] The function of these molecules is determined by their shape as well as their composition, and their structure has multiple levels. Chemical[edit] Chemical structure refers to both molecular geometry and electronic structure.

Regular numerical predicate In computer science and mathematics, more precisely in automata theory, model theory and formal language, a regular numerical predicate is a kind of relation over integers. Regular numerical predicates can also be considered as a subset of for some arity . The class of regular numerical predicate is denoted [2] and REG.[3] Definitions[edit] The class of regular numerical predicate admits a lot of equivalent definitions. and a (numerical) predicate of arity Automata with variables[edit] The first definition encodes predicate as a formal language. Let the alphabet be the set of subset of . integers , it is represented by the word of length whose -th letter is . is represented by the word We then define as The numerical predicate is said to be regular if is a regular language over the alphabet . Automata reading unary numbers[edit] This second definition is similar to the previous one. Our alphabet is the set of vectors of binary digits. . Given a length and a number , the unary representation of is the word "0"'s.

Set theory The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. History[edit] Mathematical topics typically emerge and evolve through interactions among many researchers. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation[edit] Some ontology[edit] Sets alone.

Absoluteness Mathematical logic concept Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absoluteness. In set theory, the issue of which properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences. The absoluteness of large cardinal axioms is also studied, with positive and negative results known. In model theory[edit] In model theory, there are several general results and definitions related to absoluteness. In set theory[edit] A major part of modern set theory involves the study of different models of ZF and ZFC. Other properties are not absolute: Failure of absoluteness for countability[edit] and sentences.

Zeroth-order logic First-order logic without variables or quantifiers Abstract model theory From Wikipedia, the free encyclopedia In mathematical logic, abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and their models.[1] Abstract model theory provides an approach that allows us to step back and study a wide range of logics and their relationships.[2] The starting point for the study of abstract models, which resulted in good examples was Lindström's theorem.[3] In 1974 Jon Barwise provided an axiomatization of abstract model theory.[4] See also[edit] References[edit] Further reading[edit]

Alfred North Whitehead English mathematician and philosopher Alfred North Whitehead OM FRS FBA (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy,[21] which today has found application to a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology, among other areas. In his early career Whitehead wrote primarily on mathematics, logic, and physics. Whitehead's process philosophy argues that "there is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us Life[edit] Childhood, education[edit] Whewell's Court north range at Trinity College, Cambridge. Career[edit] Bertrand Russell in 1907. Toward the end of his time in England, Whitehead turned his attention to philosophy. Move to the US, 1924[edit] God[edit]

Algebraic theory The notion is very close to the notion of algebraic structure, which, arguably, may be just a synonym. Saying that a theory is algebraic is a stronger condition than saying it is elementary. Informal interpretation[edit] An algebraic theory consists of a collection of n-ary functional terms with additional rules (axioms). This is opposed to geometric theory which involves partial functions (or binary relationships) or existential quantors − see e.g. Category-based model-theoretical interpretation[edit] proji: n → 1, i = 1, ..., n This allows interpreting n as a cartesian product of n copies of 1. Example: Let's define an algebraic theory T taking hom(n, m) to be m-tuples of polynomials of n free variables X1, ..., Xn with integer coefficients and with substitution as composition. In an algebraic theory, any morphism n → m can be described as m morphisms of signature n → 1. Note that for the case of operation 2 → 1, the appropriate algebra A will define a morphism See also[edit] References[edit]

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