# 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.

Wikimedia Foundation. 2010.

Поможем написать реферат

### Look at other dictionaries:

• Axiom of choice — This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band). In mathematics, the axiom of choice, or AC, is an axiom of set theory stating that for every family of nonempty sets there exists a family of …   Wikipedia

• Complete theory — In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent. For theories in logics which contain classical propositional logic, this is… …   Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• List of philosophy topics (I-Q) — II and thou I Ching I Ching I proposition I Thou I Thou relationshipIaIamblichus (philosopher)IbYahya Ibn Adi Yahya Ibn Adi Ibn al Arabi Muhyi al Din Ibn al Arabi Abu Bakr Ibn Bajja Abu Bakr Ibn Bājja Abu Bakr Muhammad Ibn Yahya Ibn as Say igh… …   Wikipedia

• David Makinson — David Clement Makinson, D.Phil, (born 27 August 1941), is an Australian mathematical logician living in London, England. Career Makinson began his studies at Sydney University in 1958 and was an associate of the Libertarian Society and Sydney… …   Wikipedia

• Kripke semantics — (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal… …   Wikipedia

• Model theory — This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. In mathematics, model theory is the study of (classes of) mathematical structures (e.g. groups, fields,… …   Wikipedia

• Henry E. Kyburg, Jr. — Henry E. Kyburg, Jr. (1928 – 2007) was Gideon Burbank Professor of Moral Philosophy and Professor of Computer Science at the University of Rochester, New York, and Pace Eminent Scholar at The Institute for Human and Machine Cognition, Pensacola,… …   Wikipedia

• Forcing (mathematics) — For the use of forcing in recursion theory, see Forcing (recursion theory). In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963,… …   Wikipedia

• Belief revision — is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational agents.What… …   Wikipedia