"Peano arithmetic: addition & multiplication on natural numbers is enough to be TC;" My head swims when the situation is described with this level of vagueness

"Peano arithmetic: addition & multiplication on natural numbers is enough to be TC;" My head swims when the situation is described with this level of vagueness. I mean, sure the task of proving a theorem using the modern version of the Peano postulates is undeciable and so I'd assume a map from theorems in the Peano system to proofs of theorems would be Turing complete. But a computation system based on calculating the values of simple arithmetic expressions isn't Turing complete. An express involving just adding and multiplying constant integer values will terminate.