Completeness
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In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise in various ways, several of which have a related notion of completion. It should be noted that "complete" here is just 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, compactification, or Gödel's incompleteness theorem.
- A metric space (or uniform space) is complete if every Cauchy sequence in it converges. See complete space.
- In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S. In the particular case of Hilbert spaces (or more generally, inner product spaces), an orthonormal basis is a set that is both complete and orthonormal.
- A measure space is complete if every subset of every null set is measurable. See complete measure.
- 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 if every functor from a small category to C has a limit; it is cocomplete if every such functor has a colimit. For more information, see the given article on limits in category theory.
- In order theory and related fields such as lattice and domain theory, completeness generally refers to the existence of certain suprema or 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 completeness. Up to isomorphism there is only one complete ordered field: the field of real numbers.
- In logic, a formal calculus (often just specified by a set of additional axioms used to formalize some theory within the underlying logic) is complete if, for any statement P, if <math>P</math> is true in every model of the theory, then a proof exists for P. First order logic is known to be complete. A theory is complete if it contains either <math>S</math> or <math>\neg S</math> for every sentence <math>S</math> in the language. A system is consistent if a proof never exists for both P and not P. Gödel's incompleteness theorem says that no system as powerful as the Peano axioms can be both consistent and complete. See also below for another notion of completeness in logic.
- In proof theory and related fields of mathematical logic, a formal calculus is complete with respect to a certain logic (i.e. with respect to its semantics), if every statement P that follows semantically from a set of premises G can be derived syntactically from these premises within the calculus. Formally, <math>G \models P </math> implies <math> G \vdash P </math> . Especially, all tautologies of the logic can be proven. Even when working with classical logic, this is not equivalent to the notion of completeness introduced above (both a statement and its negation might not be tautologies with respect to the logic). The reverse implication is called soundness.
- 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.