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 Let A {\displaystyle A} be a countable admissible set. Let L {\displaystyle L} be an A {\displaystyle A} -finite relational language. Suppose Γ {\displaystyle \Gamma } is a set of L A {\displaystyle L_{A}} -sentences, where Γ {\displaystyle \Gamma } is a Σ 1 {\displaystyle \Sigma _{1}} set with parameters from A {\displaystyle A} , and every A {\displaystyle A} -finite subset of Γ {\displaystyle \Gamma } is satisfiable. Then Γ {\displaystyle \Gamma } is satisfiable.
References 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.
External links Stanford Encyclopedia of Philosophy: "Infinitary Logic", Section 5, "Sublanguages of L(ω1,ω) and the Barwise Compactness Theorem"