Jack Thompson below, asks the metaphysical question whether real numbers exist, and the answer is one of definitions not of existence. No, real numbers are platonic entities. Natural numbers exist in nature. All else is a product of mind. And even natural numbers are names for the act of counting. Everything else an act of calculating ratios. 🙂
I am not sure that there are problems mathematics cannot illustrate. That seems unlikely. I think there are LOGICAL problems with number theory that cannot necessarily be solved with certainty. Math is a pretty good way of describing the physical world. It is a pretty good way of describing complex relations. But that is different from saying that math can express everything about itself. And that is probably the fundamental question that we would like to solve, but may not be able to.