Complete theory
Concept in mathematical logic
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 is complete if, for every sentence in the language, either holds or is inconsistent. Another common definition, that is equivalent if the formal system satisfies the principle of explosion, requires instead that either or its negation is provable from . Using this definition, consistency of follows automatically if "either" and "or" are read as exclusive disjunction, and it thus can be omitted from the definition. If is furthermore deductively closed, completeness reduces to the concise condition that exactly one of and is contained in for every sentence . 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.
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