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In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number. Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction.
Completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval (0,1) , which is not complete.
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions ...
Completeness (order theory), a notion that generally refers to the existence of certain suprema or infima of some partially ordered set; Complete variety, an algebraic variety that satisfies an analog of compactness; Complete orthonormal basis—see Orthonormal basis#Incomplete orthogonal sets; Complete sequence, a type of integer sequence ...
Semantic completeness is the converse of soundness for formal systems. A formal system is complete with respect to tautologousness or "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system ...
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence φ , {\displaystyle \varphi ,} the theory T {\displaystyle T} contains the sentence or its negation but not both (that is, either T ⊢ φ ...
In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers , the complex numbers , and complete valued fields (such as the p -adic numbers ).
Completeness occurs in the Lehmann–Scheffé theorem, [1] which states that if a statistic that is unbiased, complete and sufficient for some parameter θ, then it is the best mean-unbiased estimator for θ.