You can construct a formal system with any axioms you choose. It is not arbitrary that some of these systems turn out to be useful in modeling the world. But th

You can construct a formal system with any axioms you choose. It is not arbitrary that some of these systems turn out to be useful in modeling the world. But there are other systems that are only dry exercises in symbol manipulation that may be of little use to physics or engineering. Or of course they might end up being super important 100 years later. But in the meantime mathematicians might be interested in them anyway.