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.

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