In mathematics when a theory is a maximal consistent set of sentences then it is said to be complete. A maximal consistent set is a set of formulas which belongs to some formal language and that satisfies some constraints.The maximal consistent sets are said to be the primary tool in theory of classical logic and modal logic. The occurrence of the maximal consistent set is based on the idea that a disagreement involves the use of finitely many premises.
A maximal consistent set should contain the following constraints,
1) A maximal set is consistent that is there is no formula that has been proved or refused.
2) A consistent set is said to be maximal when for each formula either the set or its negation should be contained in that set.
The complete theories are said to be closed under certain conditions where it is given by the examples of the set.
For a set S: A Λ B € S if and only if A € S and also B € S
For a set S:A V B € S if and only if A € S or B € S
Proof: Let us consider the form ∆│- φ. If (¬φ) € ∆, then by ¬E we have consistency. Hence (¬φ) does not belong to ∆. By maximality φ € ∆.
Proof: Let us consider A be any atom that has been involved in a sentence of maximal consistent closed set ∆. Then determine a truth valid statement `tau` which is given by
`tau` (A) = true if A € ∆
`tau` (A) = false if A does not belongs to ∆