Maximal consistent set

Maximal consistent set

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.


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