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| | Hello, I'm Rusty, a 17 year old from Le Blanc-Mesnil, France.<br>My hobbies include (but are not limited to) Card collecting, Games Club - Dungeons and Dragons, Monopoly, Etc. and watching Two and a Half Men.<br><br>my web blog; [http://www.Tectampico.edu.mx/site/index.php?option=com_k2&view=item&id=8:speaking-in-the-monolingual-classroom-orem-ipsum-dolor-sit-amet-consectetur-adipisicing&Itemid=281 FIFA coin generator] |
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| {{Wiktionary|completeness}}
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| 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==<!-- [[Completeness (logic)]] redirects here -->
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| In [[logic]], semantic completeness is the [[Conversion (logic)|converse]] of [[soundness]] for [[formal systems]]. A formal system is "semantically complete" when all its [[tautology (logic)|tautologies]] are [[theorem]]s, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system that is consistent with the rules of the system). [[Kurt Gödel]], [[Leon Henkin]], and [[Emil Post]] all published proofs of completeness. (See [[History of the Church–Turing thesis]].) A formal system is [[consistency|consistent]] if for all formulas φ of the system, the formulas φ and ¬φ (the [[negation]] of φ) are not both theorems of the system (that is, they cannot be both proved with the rules of the system).
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| *A formal system {{mathcal|S}} is '''semantically complete''' or simply '''complete''', if every tautology of {{mathcal|S}} is a theorem of {{mathcal|S}}. That is, <math> \models_{\mathcal S} \varphi\ \to\ \vdash_{\mathcal S} \varphi</math>.<ref name="metalogic">Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971</ref>
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| *A formal system {{mathcal|S}} is '''strongly complete''' or '''complete in the strong sense''' if for every set of premises Γ, any formula which semantically follows from Γ is derivable from Γ. That is, <math> \Gamma\models_{\mathcal S} \varphi \ \to\ \Gamma \vdash_{\mathcal S} \varphi</math>.
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| *A formal system {{mathcal|S}} is '''syntactically complete''' or '''deductively complete''' or '''maximally complete''' or simply '''complete''' if for each [[Sentence (mathematical logic)|sentence]] (closed formula) φ of the language of the system either φ or ¬φ is a theorem of {{mathcal|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 propositional variable '''A''' is not a theorem, and neither is its negation, but these are not tautologies). [[Gödel's incompleteness theorem]] shows that any recursive system that is sufficiently powerful, such as [[Peano arithmetic]], cannot be both consistent and syntactically complete.
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| *A system of [[logical connective]]s is [[functional completeness|functionally complete]] if and only if it can express all [[propositional function]]s.
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| *A language is '''expressively complete''' if it can express the subject matter for which it is intended.{{Citation needed|date=December 2008}}
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| *A formal system is '''complete with respect to a property''' if and only if every sentence that has the [[property (philosophy)|property]] is a theorem.{{Citation needed|date=December 2008}}
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| ==Mathematical completeness==
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| In [[mathematics]], "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]] or [[compactification (mathematics)|compactification]].
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| * The [[completeness of the real numbers]] is one of the defining properties of the [[real number]] system. It may be described equivalently as either the completeness of '''R''' as metric space or as a partially ordered set (see below).
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| * A [[metric space]] is ''complete'' if every [[Cauchy sequence]] in it [[limit of a sequence|converges]]. See [[Complete metric space]].
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| * A [[uniform space]] is ''complete'' if every [[Cauchy net]] in it [[Net (mathematics)#Limits of nets|converges]] (or equivalently every [[Cauchy filter]] in it [[Filter (mathematics)#Convergent filter bases|converges]]).
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| * In [[functional analysis]], a [[subset]] ''S'' of a [[topological vector space]] ''V'' is ''complete'' if its [[span (linear algebra)|span]] is [[dense (topology)|dense]] in ''V''. In the particular case of [[Hilbert space]]s (or more generally, [[inner product space]]s), an [[orthonormal basis]] is a set that is both complete and [[orthonormal]].
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| * A [[measure (mathematics)|measure space]] is ''complete'' if every subset of every [[null set]] is measurable. See [[complete measure]].
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| * 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)]].
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| * More generally, any [[topological group]] can be completed at a decreasing sequence of open subgroups.
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| * In [[statistics]], a [[statistic]] is called ''complete'' if it does not allow an unbiased estimator of zero. See [[completeness (statistics)]].
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| * In [[graph theory]], a ''[[complete graph]]'' is an undirected graph in which every pair of vertices has exactly one edge connecting them.
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| * In [[category theory]], a category ''C'' is ''[[complete category|complete]]'' if every [[diagram (category theory)|diagram]] from a small category to ''C'' has a [[limit (category theory)|limit]]; it is ''[[cocomplete]]'' if every such functor has a [[colimit]].
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| * In [[order theory]] and related fields such as [[lattice (order)|lattice]] and [[domain theory]], ''[[completeness (order theory)|completeness]]'' generally refers to the existence of certain [[supremum|suprema]] or [[infimum|infima]] of some [[partially ordered set]]. Notable special usages of the term include the concepts of [[complete Boolean algebra]], [[complete lattice]], and [[complete partial order]] (cpo). Furthermore, an [[ordered field]] is ''complete'' if every non-empty subset of it that has an upper bound within the field has a [[least upper bound]] within the field, which should be compared to the (slightly different) order-theoretical notion of [[bounded complete]]ness. [[Up to]] [[isomorphism]] there is only one complete ordered field: the field of [[real number]]s (but note that this complete ordered field, which is also a lattice, is not a complete lattice).
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| * In [[algebraic geometry]], an [[algebraic variety]] is ''complete'' if it satisfies an analog of [[compact space|compactness]]. See [[complete algebraic variety]].
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| * 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.
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| == 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.
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| * In [[computational complexity theory]], a problem ''P'' is '''[[complete (complexity)|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.<br>For example, each problem in the class '''[[NP-complete]]''' is complete for the class '''[[NP (complexity)|NP]]''', under [[polynomial time|polynomial-time]], many-one reduction.
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| * 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''.
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| * In software testing, completeness has for goal the functional verification of call graph (between software item) and control graph (inside each software item).
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| * The concept of [[Completeness (knowledge bases)|completeness]] is found in [[knowledge base]] theory.
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| ==Economics, finance, and industry==
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| * [[Complete market]]s versus [[incomplete markets]]
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| * 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.
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| * Oil or gas well [[Completion (oil and gas wells)|completion]], the process of making a well ready for production.
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| == Botany ==
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| * 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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| == References ==
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| {{reflist}}
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| {{logic}}
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| [[Category:Mathematical terminology]]
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| [[Category:Mathematical logic]]
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| [[Category:Proof theory]]
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| [[Category:Metalogic]]
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| [[hr:Potpunost (razdvojba)]]
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| [[ja:完全性]]
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| [[pl:Zupełność]]
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| [[zh:完備性]]
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Hello, I'm Rusty, a 17 year old from Le Blanc-Mesnil, France.
My hobbies include (but are not limited to) Card collecting, Games Club - Dungeons and Dragons, Monopoly, Etc. and watching Two and a Half Men.
my web blog; FIFA coin generator