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Symbolic logic has been extended to a description and analysis of the foundations of mathematics, particularly number theory. Gödel also made (1931) the surprising discovery that number theory cannot be complete, i.e., that no matter what axioms are chosen as a basis for number theory, there will always be some true statements that cannot be deducted from them, although they can be proved within the larger context of symbolic logic. Since many branches of mathematics are ultimately based on number theory, this result has been interpreted by some as affirming that mathematics is an open, creative discipline whose possibilities cannot be delineated. The work of Gödel, Church, and others has led to the development of proof theory, or metamathematics, which deals with the nature of mathematics itself.

- Introduction
- Truth-functional Analysis
- The Predicate Calculus
- Analysis of the Foundations of Mathematics
- Bibliography

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.