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.
Let be a countable admissible set. Let be an -finite relational language. Suppose is a set of -sentences, where is a set with parameters from , and every -finite subset of is satisfiable. Then is satisfiable.
- 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. pp. 295. ISBN 3-540-90936-2.
- Stanford Encyclopedia of Philosophy: "Infinitary Logic", Section 5, "Sublanguages of L(ω1,ω) and the Barwise Compactness Theorem"