proof theory

References

proof theory

Also known as: mathematical proof, metamathematics

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

July 13, 2026

Quick Summary

In logic, a proof is an argument that establishes the validity of a proposition. While proofs can be based on inductive logic, the term generally implies a rigorous deduction. In formal systems of logic and mathematics, a proof is a finite sequence of formulas. Each formula is either an axiom or derived from previous formulas through valid inference, with the final formula being the one to be proven.1

Proof theory, a branch of logic, studies the purely formal aspects of logical languages and their inference rules. It focuses on the different kinds of proofs that can be conducted within formal systems. This field has been aided by the development of computers, which can assist in discovering theorems and completing proofs, especially those with a large number of cases.23

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

3 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

completeness
In proof theory, a formal system is said to be syntactically complete if and only if every closed sentence in the system is such that either it or its negation is provable in the system. In model theory, a formal system is said to be semantically…
Read More
In proof theory, a formal system is said to be syntactically complete if and only if every closed sentence in the system is such that either it or its negation is provable in the system. In model theory, a formal system is said to be semantically…
Read More

history of logic
Syntax and proof theory
The resulting “proof theory” was concerned primarily (though not exclusively) with the different kinds of proof that can be accomplished within formal systems.
Read More
The resulting “proof theory” was concerned primarily (though not exclusively) with the different kinds of proof that can be accomplished within formal systems.
Read More

metalogic
Syntax and semantics
…which is closely related to proof theory, must often be distinguished from semantics, which is closely related to model theory. Roughly speaking, syntax—as conceived in the philosophy of mathematics—is a branch of number theory, and semantics is a branch of set theory, which deals with the nature and relations of…
Read More
…which is closely related to proof theory, must often be distinguished from semantics, which is closely related to model theory. Roughly speaking, syntax—as conceived in the philosophy of mathematics—is a branch of number theory, and semantics is a branch of set theory, which deals with the nature and relations of…
Read More

logic
Logical systems
…formal system is known as proof theory. It is one of the main areas of systematic logical theory.
Read More
…formal system is known as proof theory. It is one of the main areas of systematic logical theory.
Read More

set theory
Limitations of axiomatic set theory
…axiomatic theory T is called proof theory, or metamathematics. It is premised upon the formulation of T as a formal axiomatic theory—i.e., the theory of inference (as well as T) must be axiomatized. It is then possible to present T in a purely symbolic form—i.e., as a formal language based…
Read More
…axiomatic theory T is called proof theory, or metamathematics. It is premised upon the formulation of T as a formal axiomatic theory—i.e., the theory of inference (as well as T) must be axiomatized. It is then possible to present T in a purely symbolic form—i.e., as a formal language based…
Read More