Complete theory

In mathematical logic, a theory of a language is complete if it is consistent and it proves every closed formula with which it is not inconsistent. That is to say, a consistent theory $T$ is complete if, for every sentence $\varphi$ in the language, either $T\vdash \varphi$ holds or $T\cup \{\varphi \}$ is inconsistent. Another common definition, that is equivalent if the formal system satisfies the principle of explosion, requires instead that either $\varphi$ or its negation $\neg \varphi$ is provable from $T$. Using this definition, consistency of $T$ follows automatically if "either" and "or" are read as exclusive disjunction, and it thus can be omitted from the definition. If $T$ is furthermore deductively closed, completeness reduces to the concise condition that exactly one of $\varphi$ and $\neg \varphi$ is contained in $T$ for every sentence $\varphi$. Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem.

This syntactic sense of complete is distinct from the semantic notion of a complete formal system, which asserts that for every theory that can be formulated in the formal system, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness.