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 $A$ 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.

References

{{cite thesis |type=PhD |last=Barwise |first=J. |year=1967 |title=Infinitary Logic and Admissible Sets |publisher=Stanford University}}
{{cite book |last1=Ash |first1=C. J. |last2=Knight |first2=J. |year=2000 |title=Computable Structures and the Hyperarithmetic Hierarchy |publisher=Elsevier |isbn=0-444-50072-3}}
{{cite book |last1=Barwise |first1=Jon |last2=Feferman |first2=Solomon |authorlink2=Solomon Feferman |last3=Baldwin |first3=John T. |year=1985 |title=Model-theoretic logics |publisher=Springer-Verlag |isbn=3-540-90936-2 |pages=295 |url=https://archive.org/details/modeltheoreticlo00barw/page/n314 |url-access=limited}}

External links

Stanford Encyclopedia of Philosophy: [http://plato.stanford.edu/entries/logic-infinitary/#5 "Infinitary Logic", Section 5, "Sublanguages of L(ω1,ω) and the Barwise Compactness Theorem"]

Category:Theorems in the foundations of mathematics

Category:Mathematical logic

Category:Metatheorems