On the other hand, a theorem prover does not have to be able to prove everything about a system to be able to prove some useful things. One might also say "the system mathematicians are interested in is all of mathematics, which is more powerful than arithmetic", or, for that matter, "the system programmers are interested in is all of theoretical computation, which is (at least) Turing complete"—but sometimes it's useful intentionally to restrict our power. (I think of https://news.ycombinator.com/item?id=10567408 , for example.)
(If we're being very picky, then, as pfortuny points out (https://news.ycombinator.com/item?id=12762192), a computer, having only a finite address space, isn't even as powerful as ordinary arithmetic; but it's fair to guess that this isn't a useful kind of pickiness.)
Not one that can handle Peano arithmetic—but there are plenty of interesting systems sub-Peano arithmetic. Presburger arithmetic (https://en.wikipedia.org/wiki/Presburger_arithmetic) is the one that springs to mind; I doubt that it's directly useful as a theorem prover, but it does show that interesting mathematics can be done in complete theories.