Not even sure that the small errors in the physic models are needed. The beauty of René Thom"s chaos theory is that the non-determinism lies deep within the maths.
As an example, even if you know very precisely the speed and the position of the solar's system planets, and the external perturbations, you just can't do the maths, the 3-bodies equations are already too complicated to be solved. And which planet will be ejected from the solar system is still a mystery.
You don't need to solve the equations though. In a deterministic universe, if you know
precisely the starting conditions, you could run a simulation and get the prediction. Chaos theory says that small perturbations in the starting conditions can lead to vastly different outcomes. Quantum physics says small perturbations are intrinsic. Even with two initially identical systems, things will occur in one which will not in the other.
But to run a simulation you need to be able to solve the 3-body equations... Unless you're running some sort of simplified heuristic simulation, in which case it could never be absolutely deterministic anyway.
You have to distinguish practical predictability from theoretical predictability. Chaos theory (as well as the difficulty in solving 3-body equations and such) make the universe practically unpredictable. Quantum theory (and
only quantum theory) make the universe theoretically unpredictable.
Right. You can only predict events to the extent that the underlying system is computationally reducible. Unlike the 2-body problem (that has closed-form solutions), the 3-body problem is computationally irreducible. To determine the answer to which planet gets ejected would require a computer that has more computational power than the universe itself. In other words, to find out the answer, we must observe reality unfold.
The question of whether the universe is deterministic is unresolved. Many physicists claim that wave-function collapse is real, for example citing
Bell's work to "disprove" Einstein's hidden-variable theory. But Bell only showed that no
local hidden variable theorem was possible. This means that if the universe is deterministic, there must be non-local effects. Stephen Wolfram's concept of the universe as a network of nodes allows for non-locality without violating causality.
