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 or compactification.

  • 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 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, 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 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.
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