Definition: If $A$ is a theory and $B \subseteq A$ then $B$ is a set of axioms for $A$ iff 1) B is recursive and 2) $B \models C$ for all $C \in A$. We say $A$ is axiomatizable iff $A$ has a set of axioms.
How can any theory be axiomatizable with this definition? If both theories prove the same set of sentences then wouldn't those sentences belong in both theories and $B = A$?
Definition: A theory $A$ is consistent iff it's not equal to the set of sentences in the Language of Arithmetic (with no free variables).
I thought that a consistent set is just one that doesn't have any contradictions. Does this definition relate to that? Thanks.