The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
Doesn't it though?
What happens when three bodies in a gravitationally bound system orbit each other? Our computers can't precisely compute their interaction because our computers have limited precision and discrete timesteps. Even when we discard such complicated things as relativity, what with its Lorentz factors and whatnot.
Nature can perfectly compute their interactions because it has smooth time and infinite precision.
That doesn't follow. Nature can perfectly compute them because they are nature. Nowhere is it required to have infinite precision, spatial or temporal.