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.

gollark: Apologies, my internet connection briefly incursed.

gollark: - also the cubes, I guess

gollark: Well, various possibilities exist:- add "exist again" after "exit the zzcxz" - this could take you to a new area with more doors or something- somehow add a mysterious button on the wall of one of the halls which switches the door to osmarks.net mode (somehow²)- "go in an anomalous direction" (or somehow "go up"/"go down") in the initial room- in the bee room, one series of choices could allow access to it- the "sandy ground" area after going backward

gollark: Yes, that is what I was wondering.

gollark: Hmm, where would it actually go? I guess I could repurpose one of the doors.

References

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.

External links

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

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