Recursive logic frames

by Shelah and Vaananen. [ShVa:790]
Math Logic Quarterly, 2006
We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called recursively (countably) compact, if every recursive (respectively, countable) finitely consistent theory has a model. We show that for logic frames built from the cardinality quantifiers ``there exists at least lambda'' recursive compactness always implies countable compactness. On the other hand we show that a recursively compact extension need not be countably compact.

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