complétude

Complétude, in mathematics, denotes the property that a given structure contains all necessary limit points or limits of convergent processes within itself. The precise meaning varies by context, but all standard uses share the idea that the structure is closed under a natural limiting operation. The term is commonly translated as completeness in English.

In analysis, a metric space (X, d) is complete if every Cauchy sequence in X converges to a limit in X. This guarantees that limit processes stay inside the space. Real numbers form a complete ordered field, whereas the rational numbers do not. Completeness underpins the construction of completions, such as enlarging the rationals to the reals or forming p-adic completions with respect to a p-adic metric.

In order theory, a partially ordered set is complete if every subset has a supremum (least upper bound) and an infimum (greatest lower bound); a complete lattice has these properties for all subsets. This notion ensures the existence of fixed points and well-behaved joins and meets, and it plays a central role in various areas of algebra and theoretical computer science.

In logic, a theory is complete if every sentence in its language is either provable from the axioms or its negation is provable. Gödel’s completeness theorem for first-order logic shows that semantic validity (truth in all models) implies syntactic provability, linking model theory and proof theory. Complete theories have decisive, unambiguous deductive outcomes for statements in their language.

Beyond these, complétude guides the process of completion of spaces, the study of convergent sequences and nets, and various notions of limit consistency across mathematical disciplines.