Recursive logic frames

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 complete (recursively compact, N ₀-compact), if every finite (respectively: recursive, countable) consistent theory has a model. We show that for logic frames built from the cardinality quantifiers "there exists at least λ" completeness always implies N₀-compactness. On the other hand we show that a recursively compact logic frame need not be N₀-compact.