Suppose a theory is based on $N$ independent axioms. My intuition was that any $N$ independent theorems of this theory could be chosen as the axioms, constructing the same theory. I'd like to know if it's true, but the more important question to me: Is it possible to obtain the same theory (or a broader theory containing all theorems of that theory) by less than $N$ axioms? The motivation for this question comes from here.
Edit: I understand your objections, and I agree that different numbers could be assigned to the same set of axioms. So there's no measure of brevity or irreducibility for basis of an axiomatic system?