Complete theory

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In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence ${\displaystyle \varphi ,}${\displaystyle \varphi ,} the theory ${\displaystyle T}${\displaystyle T} contains the sentence or its negation but not both (that is, either ${\displaystyle T\vdash \varphi }${\displaystyle T\vdash \varphi } or ${\displaystyle T\vdash \neg \varphi }${\displaystyle T\vdash \neg \varphi }). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first 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 of formulas ${\displaystyle S}${\displaystyle S}: ${\displaystyle A\land B\in S}${\displaystyle A\land B\in S} if and only if ${\displaystyle A\in S}${\displaystyle A\in S} and ${\displaystyle B\in S}${\displaystyle B\in S},
• For a set of formulas ${\displaystyle S}${\displaystyle S}: ${\displaystyle A\lor B\in S}${\displaystyle A\lor B\in S} if and only if ${\displaystyle A\in S}${\displaystyle A\in S} or ${\displaystyle B\in S}${\displaystyle 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.

Some examples of complete theories are:

• Presburger arithmetic
• Tarski's axioms for Euclidean geometry
• The theory of dense linear orders without endpoints
• 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
• A group of three elements
• True arithmetic or any other elementary diagram

• Lindenbaum's lemma
• Łoś–Vaught test

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