  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.
