Look up completeness in Wiktionary, the free dictionary. 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. "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, Gödel's completeness theorem or Gödel's incompleteness theorem. Wikipedia does not have an article with this exact name. ...
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Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ...
In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ...
Gödels completeness theorem is an important theorem in mathematical logic which was first proved by Kurt Gödel in 1929. ...
In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...
- 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 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, implies . First-order logic is complete in that sense. In particular, 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.
- A decision procedure (or decision algorithm, see decision problem) is complete if, whenever the answer is "yes", the algorithm finds it correctly. It is sound if every time the algorithm answers "yes", it is the correct answer.
- 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.
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