completeness

Intuitively, a logical system is complete if everything that we want can be derived in it. Thus a formalization of logic is complete if all logically valid forms of argument are derivable in the system; a system designed to codify mathematical reasoning is complete if all mathematical truths can be derived in it, and so on. Although put like this the notion seems to be entirely informal, more precise definitions can be given. A logical system is complete in the sense introduced by Gödel if and only if all valid well-formed formulae are theorems of the system. In a stronger sense a system is complete if for any well-formed formula A, either A is a theorem, or the system would become inconsistent if A were added as an axiom. See Gödel's theorem(<