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

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?

• Some hierarchical structures of mathematical concepts have been proposed, but these change as our understanding of the concepts changes. – David G. Stork Jan 21 at 4:00
• Ok, could you name some? What if we're more specific and say in ZFC or in category theory? – Anthony Jan 21 at 4:02
• In the example you gave, you are just (for a set with a binary operation) ordering by inclusion on the axioms you're imposing. This certainly is a logical thing to do, at least, within certain boundaries. I wouldn't advis e attempting to order the definition of "group" with "triangle" using a scheme like this. But I would definitely order the definitions of convex quadrilaterals this way, for example. – rschwieb Jan 21 at 15:35
• en.wikipedia.org/wiki/Arithmetical_hierarchy en.wikipedia.org/wiki/Theory_(mathematical_logic) – Anthony Jan 21 at 18:55
• If your definition of magma and monoid use the same binary symbol, you could say all magmas are monoids. In general, you can order theories by the subset relation on their instances. – PyRulez Feb 9 at 22:23