# Barwise compactness theorem

In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967.

## Statement of the theorem

Let ${\displaystyle A}$ be a countable admissible set. Let ${\displaystyle L}$ be an ${\displaystyle A}$-finite relational language. Suppose ${\displaystyle \Gamma }$ is a set of ${\displaystyle L_{A}}$-sentences, where ${\displaystyle \Gamma }$ is a ${\displaystyle \Sigma _{1}}$ set with parameters from ${\displaystyle A}$, and every ${\displaystyle A}$-finite subset of ${\displaystyle \Gamma }$ is satisfiable. Then ${\displaystyle \Gamma }$ is satisfiable.

## References

• Barwise, J. (1967). Infinitary Logic and Admissible Sets (Ph. D. Thesis). Stanford University.
• C. J. Ash; Knight, J. (2000). Computable Structures and the Hyperarithmetic Hierarchy. Elsevier. p. 366. ISBN 0-444-50072-3.
• Jon Barwise; Solomon Feferman; John T. Baldwin (1985). Model-theoretic logics. Springer-Verlag. p. 295. ISBN 3-540-90936-2.