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# Maximal consistent set

A maximal consistent set is a set of formulae belonging to some formal language that satisfy certain constraints:

• The set is consistent, that is, no formula is both provable and refutable.
• The set is closed under a number of conditions internally modelling the T-schema:
• For example, for a set $S\!$: $A \land B \in S$ iff $A \in S \land B \in S$,
• or, $(\exists x. \phi(x)) \in S$ iff $\exists t \in T. (\phi(t)) \in S)$, where T is the Herbrand universe of S.
• The set is maximal, which means that for each formula of the language, either it or its negation are in the set.

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.

03-10-2013 05:06:04