This has some interesting implications if reality is a system and we’re inside it, as I believe to be the case. First it means that we cannot ever conclusively prove whether the universe is deterministic (a la Laplace’s Demon) or random. We can still make some strong probabilistic arguments, but a full proof becomes impossible.
Intuitively, it seems like this must always be so—you can model a system, you can find a model which matches all past predictions and as time goes by it is always right (and precise and detailed) and talks about all of the (observable) actions of the systems since...but the model could be turn out to be wrong. (It just probably won’t.)
Second, it means that we can safely assume the existence of “atomic randomness” in all of our models. If the system is random, then atomic randomness is in some sense “real” and we’re done. But if the system is deterministic, then we can pretend atomic randomness is real, because the information necessary to dispel that apparent randomness is provably unavailable to us.
And yet this is surprising. Imagine you are in a room, and the lights go off. Then back on. While you might not be in a position to see the light switch/es you can infer someone switched them on and off. In order for predictions to always be bad, then it seems like some unobserved part of the system must keep changing. (And for surprises which cannot be predicted in advance, the state space must be underexplored (relative to the observer), like the lights changing color).
Intuitively, it seems like this must always be so—you can model a system, you can find a model which matches all past predictions and as time goes by it is always right (and precise and detailed) and talks about all of the (observable) actions of the systems since...but the model could be turn out to be wrong. (It just probably won’t.)
And yet this is surprising. Imagine you are in a room, and the lights go off. Then back on. While you might not be in a position to see the light switch/es you can infer someone switched them on and off. In order for predictions to always be bad, then it seems like some unobserved part of the system must keep changing. (And for surprises which cannot be predicted in advance, the state space must be underexplored (relative to the observer), like the lights changing color).