Vollständigkeitsprüfungen

Vollständigkeit (often translated as completeness) is a term used across mathematics, logic, and related disciplines to denote a condition in which nothing essential is missing and limits or implications are fully captured within a given framework. The exact meaning varies by field, but the core idea is that a system or object accounts for all relevant cases, limits, or components.

In logic, a deductive system is said to be complete if every statement that is true in every interpretation (valid) can be derived within the system. Gödel’s completeness theorem for first-order logic shows that semantic validity coincides with syntactic provability: every valid sentence is provable. This notion is distinct from Gödel’s incompleteness theorems, which concern the limits of more powerful theories.

In analysis, a metric space is complete when every Cauchy sequence converges to a limit inside the space. The real numbers are complete, while the rational numbers are not. Completeness ensures that limits of convergent processes remain within the same setting, which is essential for many proofs and constructions.

In statistics, a statistic is complete if no nontrivial function of the statistic has zero expectation for all parameter values. Completeness, together with sufficiency, underpins the Lehmann–Scheffé theorem, which guarantees the existence of unique unbiased estimators under certain conditions.

Other uses include complete lattices in order theory, where every subset has a supremum and an infimum, and complete sigma-algebras in measure theory, which contain all subsets of null sets. The concept thus provides a unifying notion of wholeness across various mathematical contexts.