In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.
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Logical completeness
In logic Logic is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic, semantic completeness is the converse Conversion is a concept in traditional logic referring to a "type of immediate inference in which from a given proposition, another proposition is inferred which has as its subject the predicate of the original proposition and as its predicate the subject of the original proposition ". The immediately inferred proposition is termed the of soundness In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word derives from the Germanic 'Sund' as in Gesundheit, for formal systems In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) one expression from one or more other expressions (premises) antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its. A formal system is "semantically complete" when all tautologies In logic, a tautology is a formula which is true in every possible interpretation. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense today are theorems In mathematics, a theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other whereas a formal system is "sound" when all theorems are tautologies. Kurt Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century,, Leon Henkin Leon Henkin was a logician at the University of California, Berkeley. He was principally known for the "Henkin completeness proof": his version of the proof of the semantic completeness of standard systems of first-order logic, and Emil Post Emil Leon Post, Ph.D., was a mathematician and logician all published proofs of completeness. (See History of the Church–Turing thesis.) A system is consistent In logic, a consistent theory is one that does not contain a 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; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term if a proof never exists for both P and not P.
- For a formal system S in formal language L, S is semantically complete or simply complete, if and only if every logically valid formula of L (every formula which is true under every interpretation of L) is a theorem of S. That is, .[1]
- A formal system S is strongly complete or complete in the strong sense if and only if for every set of premises Γ, any formula which semantically follows from Γ is derivable from Γ. That is, .
- A formal system S is syntactically complete or deductively complete or maximally complete or simply complete if and only if for each formula A of the language of the system either A or ¬A is a theorem of S. This is also called negation completeness. In another sense, a formal system is syntactically complete if and only if no unprovable axiom can be added to it as an axiom without introducing an inconsistency. Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example the propositional logic statement consisting of a single variable "a" is not a theorem, and neither is its negation, but these are not tautologies). Gödel's incompleteness theorem Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that shows that no recursive system that is sufficiently powerful, such as the Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental, can be both consistent and complete.
- A formal system is inconsistent if and only if every sentence is a theorem.[2]
- A system of logical connectives In logic, a logical connective is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences is functionally complete if and only if it can express all propositional functions.
- A language is expressively complete if it can express the subject matter for which it is intended.[citation needed]
- A formal system is complete with respect to a property if and only if every sentence that has the property In modern philosophy, mathematics, and logic, a property is an attribute of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. If, however, for every predicate there is a corresponding property, then properties are subject to Russell's is a theorem.[citation needed]
Mathematical completeness
In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, "complete" is a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F or compactification In mathematics, compactification is the process or result of enlarging a topological space to make it compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
- A metric space In mathematics, a metric space is a set where a notion of distance between elements of the set is defined (or uniform space) is complete if every Cauchy sequence In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. To be more precise, by dropping enough terms from the start of the sequence, it is possible to make the maximum of the distances from any of the remaining elements to any other such in it converges The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit. See Complete metric space In mathematical analysis, a metric space M is said to be complete if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M.
- In functional analysis Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral, a subset In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B S of a topological vector space In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space V is complete if its span is dense In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point in X can be "well-approximated" by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of x contains at least one point from A in V. If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination In mathematics, linear combinations is a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article of vectors from S. In the particular case of Hilbert spaces The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space (or more generally, inner product spaces), an orthonormal basis In mathematics, an orthonormal basis of an inner product space V , is a set of mutually orthogonal vectors of magnitude 1 (unit vectors) that span the space when infinite linear combinations are allowed. (In some contexts, especially in linear algebra, the concept of basis means a set of vectors that span a space when only finite linear is a set that is both complete and orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and both of unit length. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
- A measure space In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, volume, etcetera. A particularly important example is the Lebesgue measure on a is complete if every subset of every null set In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set . More generally, whenever an ideal is taken as understood, then a null set is any element of that ideal is measurable. See complete measure.
- In commutative algebra, a commutative ring can be completed at an ideal (in the topology defined by the powers of the ideal). See Completion (ring theory).
- More generally, any topological group In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. Topological groups allow one to study the notion of continuous symmetries in the form of continuous group actions can be completed at a decreasing sequence of open subgroups.
- In statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments, a statistic A statistic is a single measure of some attribute of a sample (e.g. its arithmetic mean value). It is calculated by applying a function (statistical algorithm) to the values of the items comprising the sample which are known together as a set of data is called complete if it does not allow an unbiased estimator of zero. See completeness (statistics) In statistics, completeness is a property of a statistic for which the statistic optimally obtains information about the unknown parameters characterizing the distribution of the underlying data.
- In graph theory In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning, a complete graph is an undirected graph in which every pair of vertices has exactly one edge connecting them.
- In category theory, a category C is complete if every diagram In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A from a small category to C has a limit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits; it is cocomplete if every such functor has a colimit.
- In order theory Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a introduction to the field and includes some of the most basic definitions. For a quick and related fields such as lattice In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum (the elements' least upper bound; called their join) and an infimum (greatest lower bound; called their meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are and domain theory Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages, completeness In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist generally refers to the existence of certain suprema In mathematics, given a subset S of a partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to each element of S. Consequently, the supremum is also referred to as the least upper bound (LUB). If the supremum exists, it may or may not belong to S. If the supremum exists, it is unique or infima In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all elements of the subset. Consequently the term greatest lower bound is also commonly used. Infima of real numbers are a common special case that is especially important in analysis. However, the general of some partially ordered set In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are. Notable special usages of the term include the concepts of complete Boolean algebra, complete lattice, and complete partial order In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. These orders, called dcpo and cpo for short, are characterized by particular completeness properties. Both dcpos and cpos are considered in domain theory and have major applications in theoretical computer science and (cpo). Furthermore, an ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations. An example of an ordered field is the field of real numbers. This concept was introduced by Emil Artin in 1927 is complete if every non-empty subset of it that has an upper bound within the field has a least upper bound In mathematics, given a subset S of a partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to each element of S. Consequently, the supremum is also referred to as the least upper bound, lub or LUB. If the supremum exists, it may or may not belong to S. If the supremum exists, it is within the field, which should be compared to the (slightly different) order-theoretical notion of bounded completeness. Up to In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent. In group theory, for example, isomorphism In abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f there is only one complete ordered field: the field of real numbers (but note that this complete ordered field, which is also a lattice, is not a complete lattice).
- In algebraic geometry, an algebraic variety is complete if it satisfies an analog of compactness. See complete algebraic variety.
- In quantum mechanics, a complete set of commuting operators (or CSCO) is one whose eigenvalues are sufficient to specify the physical state of a system.
Computing
- In algorithms, the notion of completeness refers to the ability of the algorithm to find a solution if one exists, and if not, to report that no solution is possible.
- In computational complexity theory, a problem P is complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-complete is complete for the class NP, under polynomial-time, many-one reduction.
- In computing, a data-entry field can autocomplete the entered data based on the prefix typed into the field; that capability is known as autocompletion.
- In software testing, completeness has for goal the functional verification of call graph (between software item) and control graph (inside each software item).
- The concept of completeness is found in knowledge base theory.
Economics, finance, and industry
- Complete markets versus incomplete markets
- In auditing, completeness is one of the financial statement assertions that have to be ensured. For example, auditing classes of transactions. Rental expense which includes 12-month or 52-week payments should be all booked according to the terms agreed in the tenancy agreement.
- Oil or gas well completion, the process of making a well ready for production.
Botany
- A complete flower is a flower with both male and female reproductive structures as well as petals and sepals. See Sexual reproduction in plants.
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Fri, 13 Aug 2010 22:19:47 GMT+00:00
Market Intellisearch Neither Market Intellisearch nor its affiliates warrant its completeness , accuracy or adequacy and it should not be relied upon as such. ...
Fred Karlson
ue, 11 May 2010 14:37:22 GM
A more fruitful approach is that of Jespersen (1924: 307), who suggests testing the . completeness. and independence of a sentence, by assessing its potential for standing alone, as a . complete. utterance. 1 ...
