- Metric spaces or uniform spaces are said to be
**complete**if every Cauchy sequence in them converges. See complete space. - 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. Up to isomorphism there is only one complete ordered field: the field of real numbers. - 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 lattice is
**complete**if each of its subsets has a supremum and an infimum. - 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 where every pair of vertices has exactly one edge connecting them. - In category theory, a category
*C*is called**complete**if every functor from a small category to*C*has a limit; it is called**cocomplete**if every such functor has a colimit. - In logic, a formal calculus (often just specified by a set of additional axioms used to formalize some theory within the underlying logic) is said to be
**complete**if, for any statement*P*, a proof exists for*P*or for not*P*. A system is*consistent*if a proof never exists for both*P*and not*P*. Gödel's incompleteness theorem proved 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 said to be
**complete**with respect to a certain logic (i.e. wrt its semantics), if every statement*P*, that follows sematically from a set of premises*G*, can be derived syntactically from these premisses within the calculus. Formally,*G*|=*P*implies*G*|-*P*. 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 wrt the logic). The reverse implication is called soundness. - In computational complexity theory, a problem
*P*is said to be**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.