Alexandroff extension - Revision history

en>Gausler: /* Further examples */

2014-12-23T10:32:48Z

Further examples

← Older revision		Revision as of 12:32, 23 December 2014
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173.25.54.191: del 'easily'

2014-02-25T00:45:31Z

del 'easily'

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en>JustAGal: Disambiguated: Ronald Brown → Ronald Brown (mathematician)

2013-03-22T15:58:34Z

Disambiguated: Ronald Brown → Ronald Brown (mathematician)

← Older revision		Revision as of 17:58, 22 March 2013
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	~~The title~~ of the ~~writer~~ is ~~Numbers~~. ~~For~~ a ~~while she~~'~~s been~~ in ~~South Dakota~~. He is ~~really fond~~ of ~~doing ceramics~~ but he is ~~struggling~~ to ~~find time~~ for it. ~~Hiring~~ is ~~her day job now but she~~'s ~~always needed her personal company~~.~~<br><br>Feel free~~ to ~~visit my site~~ [~~http~~:~~//www~~.~~ourabilitywiki~~.~~com/dining-~~and-~~night~~-~~life/tips~~-~~about-how~~-to-~~overcome~~-~~candida-easily home std test kit~~]		{{refimprove\|date=January 2014}}
			{{Split\|date=May 2013}}
			{{Wiktionary\|completeness}}
			In general, an object is '''complete''' if nothing needs to be added to it. This notion is made more specific in various fields.

			==Logical completeness==<!-- [[Completeness (logic)]] redirects here -->
			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).

			*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>

			*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>.

			*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.

			*A system of [[logical connective]]s is [[functional completeness\|functionally complete]] if and only if it can express all [[propositional function]]s.

			*A language is '''expressively complete''' if it can express the subject matter for which it is intended.{{Citation needed\|date=December 2008}}

			*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}}

			==Mathematical completeness==
			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]].

			* 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).

			* A [[metric space]] is ''complete'' if every [[Cauchy sequence]] in it [[limit of a sequence\|converges]]. See [[Complete metric space]].

			* 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]]).

			* 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]].

			* A [[measure (mathematics)\|measure space]] is ''complete'' if every subset of every [[null set]] 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]] can be completed at a decreasing sequence of open subgroups.

			* In [[statistics]], a [[statistic]] is called ''complete'' if it does not allow an unbiased estimator of zero. See [[completeness (statistics)]].

			* In [[graph theory]], 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 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]].

			* 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).

			* In [[algebraic geometry]], an [[algebraic variety]] is ''complete'' if it satisfies an analog of [[compact space\|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 (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.
			* 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 (knowledge bases)\|completeness]] is found in [[knowledge base]] theory.

			==Economics, finance, and industry==
			* [[Complete market]]s 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 (oil and gas wells)\|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]].

			== References ==

			{{reflist}}

			{{logic}}

			[[Category:Mathematical terminology]]
			[[Category:Mathematical logic]]
			[[Category:Proof theory]]
			[[Category:Metalogic]]

			[[hr:Potpunost (razdvojba)]]
			[[ja:完全性]]
			[[pl:Zupełność]]
			[[zh:完備性]]

en>Crasshopper: /* Example: inverse stereographic projection */

2011-12-23T07:44:29Z

Example: inverse stereographic projection

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