Complete theory

In mathematical logic, a first-order theory is complete, if for every sentence φ in its language it contains either φ itself or its negation ¬φ. Completeness for other logics with negation is defined analogously.

Recursively axiomatizable 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

Examples of complete theories are:

• Presburger arithmetic
• Tarski axioms
• 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. pp.p. 86. ISBN 978-0-412-80830-2. 

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