##  [Complete Theory](/complete-theory) 

  ##  [Complete Theory](https://mathlogic.quantumdictionary.io/complete-theory-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A first-order theory is complete if for every sentence in its language, either the sentence or its negation is entailed by the theory; equivalently, all models of the theory are elementarily equivalent and the theory decides all sentences of the language.

 

 

 

 

 





 

 



 ##  [Complete Theory](https://puremath.quantumdictionary.io/complete-theory-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A first-order theory T in a language L is complete if for every L-sentence φ either T proves φ or T proves ¬φ; equivalently, any two models of T are elementarily equivalent (T decides every sentence of the language).

 

 

 

 

 





 

 



 ##  [Complete Theory](https://algebra.quantumdictionary.io/complete-theory-2) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A theory T is complete if for every sentence φ in its language either T proves φ or T proves ¬φ; equivalently, any two models of T are elementarily equivalent, so the theory decides truth or falsity of every sentence up to logical consequence.