Maximal consistent set

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In mathematics, a maximal consistent set is a set of formulae belonging to some formal language that satisfies the following constraints:

The set is consistent, that is, no formula is both provable and refutable.
The set is maximal, which means that for each formula of the language, either it or its negation are in the set.

As a consequence, a maximal consistent set is closed under a number of conditions internally modelling the T-schema:

For a set $S\!$ : $A \land B \in S$ if and only if $A \in S$ and $B \in S$ ,
For a set $S\!$ : $A \lor B \in S$ if and only if $A \in S$ or $B \in S$

By the above properties, maximal consistent sets can be considered a canonical model for a theory T. Maximal consistent sets are a fundamental tool in the model theory of classical logic and modal logic. Their existence in a given case is usually a straightforward consequence of Zorn's lemma, based on the idea that a contradiction involves use of only finitely many premises.