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

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