For a formal system S in formal language L, S is semantically complete or simply complete, iff every logically valid formula of L (every formula which is true under every interpretation of L) is a theorem of S. That is, .[1]
A formal system S is strongly complete or complete in the strong sense iff for every set of premises T, any formula which semantically follows from T is derivable from T. That is, .
A formal system S is syntactically complete or deductively complete or maximally complete or simply complete iff for each formula A of the language of the system either A or ¬A is a theorem of S. This is also called negation completeness. In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an inconsistency. Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example the propositional logic statement consisting of a single variable "a" is not a theorem, and neither is its negation, but these are not tautologies). Gödel's incompleteness theorem shows that no recursive system that is sufficiently powerful, such as the Peano axioms, can be both consistent and complete.
A formal system is extremely complete or complete in the extreme sense iff every sentence is a theorem. Not all formal systems are intended to express "truths" or "falsehoods," however for those that do an extremely complete formal system is the same as an inconsistent formal system.
A language is expressively complete if it can express the subject matter for which it is intended.[citation needed]
A formal system is complete with respect to a property iff every sentence that has the property is a theorem.[citation needed]
Mathematical completeness
In mathematics, "complete" is a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field or compactification.
In algorithms, the notion of completeness refers to the ability of the algorithm to find a solution if one exists, and if not, reports that no solution is possible.
In computational complexity theory, a problem P is complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-complete is complete for the class NP, under polynomial-time, many-one reduction.
In computing, a data-entry field can autocomplete the entered data based on the prefix typed into the field; that capability is known as autocompletion.
In software testing, completeness has for goal the functional verification of call graph (between software item) and control graph (inside each software item).
In auditing, completeness is one of the financial statement assertions that have to be ensured. For example, auditing classes of transactions. Rental expense which includes 12-month or 52-week payments should be all booked according to the terms agreed in the tenancy agreement.
Oil or gas well completion, the process of making a well ready for production.
References
^Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971