### 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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