While the space of possible axioms is infinite, and thus the possible mathematics you can derive from axioms is also potentially infinite, I think that a lot of

While the space of possible axioms is infinite, and thus the possible mathematics you can derive from axioms is also potentially infinite, I think that a lot of that space is self-isomorphic, so it is possible that in actuality there are a finite set of mathematical relationships. To put it another way, the possibility space of mathematical systems could be like a mosaic made from a finite set of interlocking tiles. Fundamentally, there has to be a layer of reality with no fixed rules. If the rules are a product of the system then they are affected by that system and it is almost certain that they change at some reachable points in the system's state space. If the rules are not a product of the system, then by definition there is a reality outside that system. That reality outside the system then suffers the same issue - either it has no fixed rules, or it is itself contained.