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Theory (mathematical Logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a deductively closed theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subseteq T that is called "the set of axioms" of the theory T, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language) When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Model (model Theory)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a ''semantic model'' when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as "interpretations", whereas the term "interpretation" generally h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hilbert-style Deductive System
:''In mathematical physics, ''Hilbert system'' is an infrequently used term for a physical system described by a C*-algebra.'' In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob FregeMáté & Ruzsa 1997:129 and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference. Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemes. The most commonly studied Hilbert systems have either just one rule of inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Expansion (model Theory)
In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The opposite of "reduct" is "expansion." Definition Let ''A'' be an algebraic structure (in the sense of universal algebra) or a structure in the sense of model theory, organized as a set ''X'' together with an indexed family of operations and relations φ''i'' on that set, with index set ''I''. Then the reduct of ''A'' defined by a subset ''J'' of ''I'' is the structure consisting of the set ''X'' and ''J''-indexed family of operations and relations whose ''j''-th operation or relation for ''j'' ∈ ''J'' is the ''j''-th operation or relation of ''A''. That is, this reduct is the structure ''A'' with the omission of those operations and relations φ''i'' for which ''i'' is not in ''J''. A structure ''A'' is an expansion of ''B'' just when ''B'' is a reduct of ''A''. That is, reduct and expansion are mutual converses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic. Definition Formally, a (single-sorted) signature can be defined as a 4-tuple , where ''S''func and ''S''rel are disjoint sets not containing any other basic logical symbols, called respectively * ''function symbols'' (examples: +, ×, 0, 1), * ''relation symbols'' or ''predicates'' (examples: ≤, ∈), * ''constant symbols'' (examples: 0, 1), and a function ar: ''S''func \cup ''S''rel → \mathbb N which assigns a natural number called ''arity'' to every function or relation symbol. A function or relation symbol is called ''n''-ary if its arity is ''n''. Some authors define a nullary (0-ary) function symbol as ''constant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Structure (mathematical Logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a '' semantic model'' when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as "interpretations", whereas the term "interpretation" generally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Many-to-one
In systems analysis, a one-to-many relationship is a type of cardinality that refers to the relationship between two entities (see also entity–relationship model) A and B in which an element of A may be linked to many elements of B, but a member of B is linked to only one element of A. For instance, think of A as books, and B as pages. A book can have many pages, but a page can only be in one book. In a relational database, a one-to-many relationship exists when one row in table A may be linked with many rows in table B, but one row in table B is linked to only one row in table A. It is important to note that a one-to-many relationship is not a property of the data, but rather of the relationship itself. A list of authors and their books may happen to describe books with only one author, in which case one row of the books table will refer to only one row of the authors table, but the relationship itself is not one-to-many, because books ''may'' have more than one author, forming ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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ω-consistent Theory
In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative) W. V. O. Quine (1971), ''Set Theory and Its Logic''. theory is a theory (collection of sentences) that is not only (syntactically) consistent (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem. Definition A theory ''T'' is said to interpret the language of arithmetic if there is a translation of formulas of arithmetic into the language of ''T'' so that ''T'' is able to prove the basic axioms of the natural numbers under this translation. A ''T'' that interprets arithmetic is ω-inconsistent if, for some property ''P'' of natural numbers (defined by a formula in the language of ''T''), ''T'' proves ''P''(0), ''P''(1), ''P''(2), and so on (that is, for every standard natural numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complete Theory
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or its negation but not both (that is, either T \vdash \varphi or T \vdash \neg \varphi). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of ''complete'' is distinct from the notion of a complete ''logic'', which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. Complete theories are closed under a number of conditions internally modelling the T-schema: * For a set of for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |