Where does completeness collide?

In mathematical analysis, a metric space is complete when every Cauchy sequence in it converges to a point that also lies in the space, so the space has no "holes" where a limit ought to be; the rationals are incomplete, and the real numbers are their completion. Fréchet's 1906 Sur quelques points du calcul fonctionnel introduced the abstract metric setting in which this property is defined, and Rudin's Principles of Mathematical Analysis gives the standard modern treatment. The referent is a structural property of a space, with nothing in common with a logic's proof power or a statistic's informational sufficiency beyond the shared word.[1][2]

Unit

Whether every Cauchy sequence in the space converges to a point within it

Recovery means

Not applicable

Measured by

Established by proving every Cauchy sequence has a limit in the space, or by constructing the space as the completion of a dense subspace

In mathematical logic, a deductive system is complete when every statement that is true in all models (every logical validity) can actually be derived from its axioms and rules, so provability matches semantic consequence. Gödel's 1930 completeness theorem, Die Vollständigkeit der Axiome des logischen Funktionenkalküls, established exactly this for first-order predicate logic. The referent is a match between a proof calculus and logical truth, distinct from (and usually paired with) soundness, and not to be confused with Gödel's later incompleteness results about arithmetic.[3]

Unit

Whether every semantically valid formula is also derivable in the system

Recovery means

Not applicable

Measured by

Established by a completeness metatheorem showing every valid sentence is provable

In statistics, a statistic is complete for a family of distributions when no non-trivial function of it has expected value zero across the whole family; intuitively, the statistic carries no "estimable" information that averages away to nothing for every parameter value. Lehmann and Casella's Theory of Point Estimation develops the notion, which together with sufficiency yields the Lehmann–Scheffé route to the best unbiased estimator. The referent is an informational property of an estimator, unrelated to a space's limit points or a logic's proof power.[4]

Unit

Whether only the zero function of the statistic has zero expectation at every parameter value

Recovery means

Not applicable

Measured by

Verified by showing that E[g(T)] = 0 for all parameter values forces g to be zero almost everywhere