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.

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.

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

• Stanford Encyclopedia of Philosophy: "Infinitary Logic", Section 5, "Sublanguages of L(ω1,ω) and the Barwise Compactness Theorem"

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