Science Fair Project Encyclopedia
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 : iff ,
- or, iff , 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.
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