I strongly agree on the statement, that the problem is undecidable. The axiom is that the machine that predicts the future exists, so then we can use that as a lever. Build another machine that predicts predictor (antipredictor). Then you can prove it by observing antipredictor. If antipredictor lights up, you dont press the button (because you want to fool predictor) but then the future in which you pressed predictor does not exists (because antipredictor lighting up automatically means you must have pressed it!). Essentially, creating new devices that predict the future might as well create different systems from which you wont be able to observe other systems. In fact the whole dispute is pointless, even if our lives are deterministic, you wont be able to see it. In our world we can only use heuristics to determine the future of systems, but complete simulation of the future is impossible, because we would need to account for everything (and use this everything to compute everything). You can only build such machines / deduct such observations when you are outside of the system.
We can deduct another experiment. Take 2D world in which 2D creatures live. For them 3D space is space-time. For us it's 4D. Then 4D creatures can deduct that our world is deterministic or not. But 4D creatures cannot determine if their space-time is deterministic. 5D creatures can say that. It goes to infinity, if at least one space-time is non-deterministic then whole system is non-deterministic. The fact that this experiment can go into the infinity makes subject undecidable.
We can deduct another experiment. Take 2D world in which 2D creatures live. For them 3D space is space-time. For us it's 4D. Then 4D creatures can deduct that our world is deterministic or not. But 4D creatures cannot determine if their space-time is deterministic. 5D creatures can say that. It goes to infinity, if at least one space-time is non-deterministic then whole system is non-deterministic. The fact that this experiment can go into the infinity makes subject undecidable.