- Source: 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
{\displaystyle A}
be a countable admissible set. Let
L
{\displaystyle L}
be an
A
{\displaystyle A}
-finite relational language. Suppose
Γ
{\displaystyle \Gamma }
is a set of
L
A
{\displaystyle L_{A}}
-sentences, where
Γ
{\displaystyle \Gamma }
is a
Σ
1
{\displaystyle \Sigma _{1}}
set with parameters from
A
{\displaystyle A}
, and every
A
{\displaystyle A}
-finite subset of
Γ
{\displaystyle \Gamma }
is satisfiable. Then
Γ
{\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"