Completeness

Completeness is the quality of having all required or possible elements, components, or information. In everyday use it denotes wholeness or finishing a task. In formal domains, completeness is a technical property that can take different precise meanings depending on the field, from mathematics to data quality and logic.

In mathematics, completeness has several precise senses. A metric space is complete if every Cauchy sequence converges to a point in the space; the real numbers form a complete ordered field, while the rationals do not. Dedekind completeness concerns the existence of least upper bounds for all bounded above subsets.

In logic, completeness usually means that every semantically true sentence is syntactically provable. Gödel's completeness theorem applies to first-order logic: if a sentence is true in every model, it is derivable from the axioms. This is distinct from notions of truth in specific, restricted models.

In statistics, completeness can refer to data quality or to a property of estimators. A complete statistic contains all information about a parameter (the Lehmann–Scheffé result). In practice, complete data means no missing values; missing data may be addressed by imputation or by complete-case analysis, each with trade-offs.

In computer science, completeness marks maximal expressiveness within a class. A problem is NP-complete if it is in NP and every NP problem reduces to it, indicating among the hardest problems in NP. A set of logic connectives is functionally complete if it can express all truth functions.

In research and data systems, completeness concerns coverage, documentation, and transparency. Checklists aim to ensure reporting includes all essential elements. Assessing completeness guides data quality, audits, and reproducibility by identifying missing information or unspecified analysis boundaries.