HOME
TheInfoList



In
mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...
, a
theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research ...
is complete if, for every closed formula in the theory's language, that formula or its
negation In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argume ...
is demonstrable. 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 S: A \land B \in S if and only if A \in S and B \in S, *For a set of formulas 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of
classical logicClassical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world. Chara ...
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 order, dense linear orders without endpoints * The theory of algebraically closed fields of a given Characteristic (algebra), characteristic * The theory of real closed fields * Every Morley's categoricity theorem, uncountably categorical Countable set, countable theory * Every omega-categorical theory, countably categorical countable theory *
group of three elements


See also

*Lindenbaum's lemma *Łoś–Vaught test


References

* Mathematical logic Model theory {{mathlogic-stub