Does anyone ever proved that by introducing new compatible axioms we always narrow down the set of possible valid deductions? I think this is intuitively correct (but the fact it is false may in fact be counterintuitive).
So given some axioms we prove theorem $\Bbb P$.
Adding further axioms can't increase the number of cases where we can apply the theorem. So if $\Bbb P$ is our result, adding new axioms to our "language" just reduce the result to something smaller.
If I understand correctly most times people try in fact to remove as much axioms as possible from the "language" and also to remove as much hypothesis as possible from theorems to make them applicable in most cases.