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 $\mathrm{\Gamma }$ is a set of $L_A$-sentences, where $\mathrm{\Gamma }$ is a ${\mathrm{\Sigma }}_1$ set with parameters from $A$, and every $A$-finite subset of $\mathrm{\Gamma }$ is satisfiable. Then $\mathrm{\Gamma }$ 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. pp. 295. ISBN 3-540-90936-2. https://archive.org/details/modeltheoreticlo00barw/page/n314. 

External links

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