I am not a mathematician, so forgive me if this question is trivial. The basic idea of my question is: For a given set of provable statements, can we find an axiom system with the smallest number of true statements (smallest by inclusion)? So here we go with my question.

Suppose we have a set of axioms A and a logic system L that we use to prove theorems arising from A. Let P(A) be the set of provable statements from A using L. Let S be the set of all other sets axiom systems B for which P(A) is a subset of P(B).

Now for any axiom system B in S, let T(B) be the set of all true statements arising from B and L. Partially order the set of axioms in S such that for B1, B2 in S, B1 < B2 if T(B1) is a subset of T(B2).

Question: Does there exist an axiom system M in S such that M < B for all B in S?