Maximal consistent set

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A maximal consistent set is a set of formulae belonging to some formal language that satisfy 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 example, 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
  • or, (\exists x. \phi(x)) \in S if and only if \exists t \in T. (\phi(t)) \in S), where T is the Herbrand universe of 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.

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