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 [math]\displaystyle{ A }[/math] be a countable admissible set. Let [math]\displaystyle{ L }[/math] be an [math]\displaystyle{ A }[/math]-finite relational language. Suppose [math]\displaystyle{ \Gamma }[/math] is a set of [math]\displaystyle{ L_A }[/math]-sentences, where [math]\displaystyle{ \Gamma }[/math] is a [math]\displaystyle{ \Sigma_1 }[/math] set with parameters from [math]\displaystyle{ A }[/math], and every [math]\displaystyle{ A }[/math]-finite subset of [math]\displaystyle{ \Gamma }[/math] is satisfiable. Then [math]\displaystyle{ \Gamma }[/math] 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"
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