Is there such a thing as ordering algebraic theories and, secondarily, their theorems?

Question

Often, after learning a new definition, I find myself wondering what the "simplest" thing I can say now is, and the next "simplest" and so on. I do the same for structures as well. It seems like it must be possible to design measures on these things and partially order them. Like to me, obviously magma $ \ \le \ $ monoid due to what is required to express either (in standard foundations I guess). Is the arithmetic hierarchy something that can be used for this, both for theorems and for theories? If not, is there such a tool or project already? In either case, where can I read more about this?