Completeness

Mathematical concept

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Completeness means that in mathematics and related fields, when an object has completeness, that is, it does not need to add any other elements, it can also be called complete or completely Of. Completeness is also called completeness, which can be accurately described from many different angles definition At the same time, we can introduce the completion concept 。

Chinese name: Completeness
Foreign name: uniform space

Also called: Completeness
Origin: Godel's incomplete definition

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Meaning in different fields

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In different fields, "completion" has different meanings. Especially in some fields, the process of "completion" is not called "completion". For other expressions, please refer to Algebraic closed field （algebraically closed field）、 Compactification (compaction) or Godel Incomplete theorem. Completeness in a general space means that the uniform convergence limit of Cauchy point sequence in any space is contained in this space. Completeness is related to the defined metric. Once the metric is defined, the completeness of this space can be discussed.^[1]

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metric space

One metric space or Uniform space (uniform space) is called "complete" Cauchy Liedu convergence (conversions), see Complete space 。^[2]

stay functional analysis (functional analysis), a topology vector space (topological vector space) The subset S of V is said to be complete, if the extension (span) of S in V is dense (deny). If V is separable topological space (separable topology space), it can also be derived that any vector in V can be written as a linear combination (finite or infinite) of elements in S. More specifically, in Hilbert space (Hilbert space) Inner product space (inner product space), a group of Orthonormal basis (orthonormal basis) is a complete and orthogonal set.^[3]

Measure space

One Measure space (measure space) is complete, if any of its Zero measure set Any subset of (null set) is Measurable Of. Please check the full measure Complete measure.^[4]

statistics

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In statistics, a statistic (statistical) is called complete, if it does not allow 0 Unbiased estimator （estimator）。 Please check the complete statistics.

graph theory

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stay graph theory (graph theory), a chart It is called complete graph. If the graph is Undirected graph , and any two vertex There is just one edge between them.

Categorical theory

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stay Categorical theory (category theory), one category C is called complete if any of the Subcategory All functors to C have limit (limit）。 And it is called upper complete if any functor has a Upper limit （colimit）。 See the definition of limit in category theory.

Order theory

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stay Order theory (order theory) and related fields, such as grid (lattice) and domain theory, Total order Completeness generally refers to Poset (Partially ordered set) A specific Supremum (suprema) or Infimum (infima）。 It is worth noting that this concept can also be applied to complete Boolean algebra (complete Boolean algebra）， Perfect lattice (complete lattice) and Complete partial order （complete partial order）。 And an ordered field is called complete if any of its Nonempty subset , there is a least upper bound in this domain; Note that this definition is related to Order theory Complete in Boundedness (bounded complete) There are minor differences. stay isomorphism In the sense of real number 。

mathematical logic

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stay mathematical logic (en: in mathematical logic), a theory is called complete. If for any sentence S in its language, the theory includes and only includes S or the inverse of S. A system is compatible if there is no proof of P and non P at the same time. Godel Incomplete Theorem Proved, including Peano axioms (Peano axioms) Axiomatic system It is impossible to be both complete and compatible. There are also some definitions of completeness in the logic below.

stay Proof theory (proof theory) and related fields of mathematical logic calculus (calculus) is complete relative to a specific logic (that is, relative to its semantics). If any statement P derived from the semantics of a set of premises Q can be derived syntactically from this set of previous proposals using this calculus syntax. Formally, Q ╞ P leads to Q | - P. First-order logic (First order logic) is complete in this sense. In particular, all logical Tautology (tautology) can be proved. Even when Classical logic This is different from the completeness of the foregoing (that is, a statement and negative statement cannot be tautologies for this logic). The opposite concept is called reliability （soundness）。

Complexity

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stay Calculation complexity In the computational complexity theory, a problem P for a complexity class C reduction The following is complete. If P is in C, any problem in C can be reduced to P. For example, NP complete problem (NP complete) in NP (NP) class and polynomial time (polynomial time) and many to one reduction are complete.

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