Above I said the only reasonable way to weaken Peano arithmetic was to weaken induction. But that's not quite true. There's another way: Not changing any of the axioms, but weakening the underlying logic.
See, there's a school of mathematics known as "constructivism", and they object to the usual laws of logic, saying that they're too strong; they use a weaker logic. It ought to be called "constructive logic", but for historical reasons (that are really not worth getting into) it's called "intuitionistic logic" instead.
And this might seem to be a better approach, because weakening the laws of logic is something that can be done without regard for the setting; it doesn't matter here whether you're talking about just whole numbers, or mathematics as a whole. (Well, almost. Actually, from ZFC and intuitionistic logic, you can effectively get back classical logic. But there are pretty good ideas about how to modify ZFC to avoid that.)
The problem is, doing this doesn't help. Because it's known that if you can prove a contradiction from the Peano axioms using classical logic, you can also do so using intuitionistic logic. So even if you went constructive, you'd still be stuck with all the same problems.