Complete theory

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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 equivalent to asking that for every sentence φ in the language of the theory it contains either φ itself or its negation ¬φ.

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.

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.

Complete theories are 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 $B\in S$ .

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 model of T, called the canonical model.

Examples

Some examples of complete theories are:

Presburger arithmetic
Tarski's axioms for Euclidean geometry
The theory of dense linear orders
The theory of algebraically closed fields of a given characteristic
The theory of real closed fields
Every uncountably categorical countable theory
Every countably categorical countable theory

References

Mendelson, Elliott (1997). Introduction to Mathematical Logic (Fourth edition ed.). Chapman & Hall. p. 86. ISBN 978-0-412-80830-2.

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& ∨ ¬ ~ → ⊃ ≡ \| ∀ ∃ ⊤ ⊥ ⊢ ⊨ ∴ ∵

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