Barwise compactness theorem

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 {displaystyle A} be a countable admissible set. Let {displaystyle L} be an {displaystyle A} -finite relational language. Suppose {displaystyle Gamma } is a set of {displaystyle L_{A}} -sentences, where {displaystyle Gamma } is a {displaystyle Sigma _{1}} set with parameters from {displaystyle A} , and every {displaystyle A} -finite subset of {displaystyle Gamma } is satisfiable. Then {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" This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.

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