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| In [[mathematical logic]], a [[Theory (mathematical logic)|theory]] is '''complete''' if it is a '''maximal consistent set''' of sentences, i.e., if it is [[consistency|consistent]], and none of its proper extensions is consistent. For theories in logics which contain [[classical logic|classical propositional logic]], this is equivalent to asking that for every [[sentence (mathematical logic)|sentence]] φ in the [[formal language|language]] of the theory it contains either φ itself or its negation ¬φ.
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| Recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by [[Gödel's incompleteness theorem]].
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| This sense of ''complete'' is distinct from the notion of a complete ''logic'', which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). [[Gödel's completeness theorem]] is about this latter kind of completeness.
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| Complete theories are closed under a number of conditions internally modelling the [[T-schema]]:
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| *For a set <math>S\!</math>: <math>A \land B \in S</math> if and only if <math>A \in S</math> and <math>B \in S</math>,
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| *For a set <math>S\!</math>: <math>A \lor B \in S</math> if and only if <math>A \in S</math> or <math>B \in S</math>.
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| 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. In the case of modal logics, the collection of maximal consistent sets extending a theory ''T'' (closed under the necessitation rule) can be given the structure of a [[Kripke semantics|model]] of ''T'', called the canonical model.
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| ==Examples== | |
| Some examples of complete theories are:
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| * [[Presburger arithmetic]]
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| * [[Tarski's axioms]] for [[Euclidean geometry]]
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| * The theory of [[dense linear order]]s
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| * The theory of [[algebraically closed field]]s of a given characteristic
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| * The theory of [[real closed field]]s
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| * Every [[Morley's categoricity theorem|uncountably categorical]] countable theory
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| * Every [[omega-categorical theory|countably categorical]] countable theory
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| ==References==
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| {{Portal|Logic}}
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| * {{cite book |first=Elliott |last=Mendelson |title=Introduction to Mathematical Logic |edition=Fourth edition |year=1997 |publisher=Chapman & Hall |isbn=978-0-412-80830-2| pages=86}}
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| {{Logic}}
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| [[Category:Mathematical logic]]
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| [[Category:Model theory]]
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| {{mathlogic-stub}}
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The title of the author is Nestor. Delaware is our beginning location. The favorite hobby for my kids and me is dancing and now I'm attempting to make cash with it. His working day job is a cashier and his salary has been truly fulfilling.
Here is my blog post: dutchtaskforce.com