Complete theory

Concept in mathematical logic

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (January 2025) (Learn how and when to remove this message)

|

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 φ , {\displaystyle \varphi ,} the theory  T {\displaystyle T} contains the sentence or its negation but not both (that is, either  T ⊢ φ {\displaystyle T\vdash \varphi } or  T ⊢ ¬ φ {\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  S {\displaystyle S} :  A ∧ B ∈ S {\displaystyle A\land B\in S} if and only if  A ∈ S {\displaystyle A\in S} and  B ∈ S {\displaystyle B\in S} ,
* For a set of formulas  S {\displaystyle S} :  A ∨ B ∈ S {\displaystyle A\lor B\in S} if and only if  A ∈ S {\displaystyle A\in S} or  B ∈ 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.

## Examples

[edit]

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

## See also

[edit]

* Philosophy portal

* Lindenbaum's lemma
* Łoś–Vaught test

## References

[edit]

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

* v
* t
* e

Mathematical logic

General|

* Axiom
* list
* Cardinality
* First-order logic
* Formal proof
* Formal semantics
* Foundations of mathematics
* Information theory
* Lemma
* Logical consequence
* Model
* Theorem
* Theory
* Type theory

Theorems (list)
and paradoxes|

* Gödel's completeness and incompleteness theorems
* Tarski's undefinability
* Banach–Tarski paradox
* Cantor's theorem, paradox and diagonal argument
* Compactness
* Halting problem
* Lindström's
* Löwenheim–Skolem
* Russell's paradox

Logics|

| Traditional|

* Classical logic
* Logical truth
* Tautology
* Proposition
* Inference
* Logical equivalence
* Consistency
* Equiconsistency
* Argument
* Soundness
* Validity
* Syllogism
* Square of opposition
* Venn diagram

|

Propositional|

* Boolean algebra
* Boolean functions
* Logical connectives
* Propositional calculus
* Propositional formula
* Truth tables
* Many-valued logic
* 3
* finite
* ∞

Predicate|

* First-order
* list
* Second-order
* Monadic
* Higher-order
* Fixed-point
* Free
* Quantifiers
* Predicate
* Monadic predicate calculus

Set theory|

* Set
* hereditary
* Class
* (Ur-)Element
* Ordinal number
* Extensionality
* Forcing
* Relation
* equivalence
* partition
* Set operations:
* intersection
* union
* complement
* Cartesian product
* power set
* identities

Types of sets|

* Countable
* Uncountable
* Empty
* Inhabited
* Singleton
* Finite
* Infinite
* Transitive
* Ultrafilter
* Recursive
* Fuzzy
* Universal
* Universe
* constructible
* Grothendieck
* Von Neumann

Maps and cardinality|

* Function/Map
* domain
* codomain
* image
* In/Sur/Bi-jection
* Schröder–Bernstein theorem
* Isomorphism
* Gödel numbering
* Enumeration
* Large cardinal
* inaccessible
* Aleph number
* Operation
* binary

Set theories|

* Zermelo–Fraenkel
* axiom of choice
* continuum hypothesis
* General
* Kripke–Platek
* Morse–Kelley
* Naive
* New Foundations
* Tarski–Grothendieck
* Von Neumann–Bernays–Gödel
* Ackermann
* Constructive

Formal systems (list),
language and syntax|

* Alphabet
* Arity
* Automata
* Axiom schema
* Expression
* ground
* Extension
* by definition
* conservative
* Relation
* Formation rule
* Grammar
* Formula
* atomic
* closed
* ground
* open
* Free/bound variable
* Language
* Metalanguage
* Logical connective
* ¬
* ∨
* ∧
* →
* ↔
* =
* Predicate
* functional
* variable
* propositional variable
* Proof
* Quantifier
* ∃
* !
* ∀
* rank
* Sentence
* atomic
* spectrum
* Signature
* String
* Substitution
* Symbol
* function
* logical/constant
* non-logical
* variable
* Term
* Theory
* list

Example axiomatic
systems (list)|

* of arithmetic:
* Peano
* second-order
* elementary function
* primitive recursive
* Robinson
* Skolem
* of the real numbers
* Tarski's axiomatization
* of Boolean algebras
* canonical
* minimal axioms
* of geometry:
* Euclidean:
* Elements
* Hilbert's
* Tarski's
* non-Euclidean

* Principia Mathematica

Proof theory|

* Formal proof
* Natural deduction
* Logical consequence
* Rule of inference
* Sequent calculus
* Theorem
* Systems
* axiomatic
* deductive
* Hilbert
* list
* Complete theory
* Independence (from ZFC)
* Proof of impossibility
* Ordinal analysis
* Reverse mathematics
* Self-verifying theories

Model theory|

* Interpretation
* function
* of models
* Model
* equivalence
* finite
* saturated
* spectrum
* submodel
* Non-standard model
* of arithmetic
* Diagram
* elementary
* Categorical theory
* Model complete theory
* Satisfiability
* Semantics of logic
* Strength
* Theories of truth
* semantic
* Tarski's
* Kripke's
* T-schema
* Transfer principle
* Truth predicate
* Truth value
* Type
* Ultraproduct
* Validity

Computability theory|

* Church encoding
* Church–Turing thesis
* Computably enumerable
* Computable function
* Computable set
* Decision problem
* decidable
* undecidable
* P
* NP
* P versus NP problem
* Kolmogorov complexity
* Lambda calculus
* Primitive recursive function
* Recursion
* Recursive set
* Turing machine
* Type theory

Related|

* Abstract logic
* Algebraic logic
* Automated theorem proving
* Category theory
* Concrete/Abstract category
* Category of sets
* History of logic
* History of mathematical logic
* timeline
* Logicism
* Mathematical object
* Philosophy of mathematics
* Supertask

Mathematics portal

* v
* t
* e

Logic

* History

Major fields|

* Computer science
* Formal semantics (natural language)
* Inference
* Philosophy of logic
* Proof
* Semantics of logic
* Syntax

| Logics|

* Classical
* Informal
* Critical thinking
* Reason
* Mathematical
* Non-classical
* Philosophical

|

Theories|

* Argumentation
* Metalogic
* Metamathematics
* Set

Foundations|

* Abduction
* Analytic and synthetic propositions
* Antecedent
* Consequent
* Contradiction
* Paradox
* Antinomy
* Deduction
* Deductive closure
* Definition
* Description
* Dichotomy
* Entailment
* Linguistic
* Form
* Induction
* Logical truth
* Name
* Necessity and sufficiency
* Premise
* Probability
* Proposition
* Reference
* Statement
* Substitution
* Truth
* Validity

Lists|

| Topics|

* Mathematical logic
* Boolean algebra
* Set theory

|

Other|

* Logicians
* Rules of inference
* Paradoxes
* Fallacies
* Logic symbols

* Category
* Outline
* Portal
* WikiProject
* changes

This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.

|

* v
* t
* e

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template