Barwise compactness theorem

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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 of the theorem[edit]

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.


  • 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. p. 295. ISBN 3-540-90936-2. 

External links[edit]

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