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.
Best Answer
1) Note that a set of sentences is not the same as the set of sentences that can be derived from the given set oif sentences.
2) Once your theory contains any contradiction, you can prove anything (ex falso quodlibet - with mild assumptions about the allowed rules of inference)