In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.

In logic, semantic completeness is the converse of soundness for formal systems. A formal system is "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system that is consistent with the rules of the system). Kurt Gödel, Leon Henkin, and Emil Post all published proofs of completeness. (See History of the Church–Turing thesis.) A formal system is consistent if for all formulas φ of the system, the formulas φ and ¬φ (the negation of φ) are not both theorems of the system (that is, they cannot be both proved with the rules of the system).

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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.