This would be set theory  rather than `ordinary math'. Still, it's interesting to observe that without Unbounded  Separation many of the customarily equivalent formulations of finiteness diverge.
For example, it is no longer the case that the system of Zermelo naturals and the system of von  Neumann naturals are isomorphic. This is discussed in "Natural Number Arithmetic in the Theory of Finite Sets" by Mayberry-Pettigrew, http://arxiv.org/abs/0711.2922.