I mean here a logic in the sense of a language and semantics. By strongly axiomatizable I mean strongly sound and strongly complete. So I'm basically asking if there is a particular deductive system for which strong completeness and soundness hold, but compactness fails.
Obviously the deductive system cannot be finitary, for if it were then compactness would follow from completeness. But does it follow immediately that a deductive system containing an infinitary rule prevents compactness? I can see that it must but don't see how to 'prove' it, without using induction up to at least the ordinal associated with the infinitary rule.
Hence, a sub-question: Is there a standard way to 'prove' such a result?
For motivation, I know for example that infinitary logics are weakly complete and not compact.