One of the major pillars of measure theory is countable subadditivity, which serves to allow one to work with countably infinite series in a nice way. This is p

One of the major pillars of measure theory is countable subadditivity, which serves to allow one to work with countably infinite series in a nice way. This is pretty much unworkable in a countable space like the computables. That isn't to say we really "need" the reals, merely that math hasn't come up with a fully-featured alternative framework that physics can use. I know there are people working on "computable measure theory" in recent years, and there's no reason why the universe should use a mathematical framework that is easy for we humans to construct.