I have seen quite a few times the assertion "Norton's dome demonstrates non-determinism of ***" (replace *** by "Newtonian mechanics" or something else), cf. e.g. this Phys.SE post.
I know that in such an example, the differential equation of the trajectory of a point left to gravity on that dome has strictly more than one solution. However, I don't understand what this has to do with determinism. Let me explain myself.
Metaphysically, I think that the purpose of a physical theory is to predict that some stories (describing events) cannot happen. For example, the weakest theory says: "anything can happen". I think that such a theory should not be called nondeterministic, but just incomplete. Some theories, like Newtonian mechanics for non-degenerate cases, are of the form "only one story can happen, given such initial condition". I think we should call this kind of theories complete theories.
Moreover, should determinism/indeterminism be really a feature of a theory, of the thing it is supposed to model?
Assume we want to describe the process of a machine programmed to extract a ball from an urn, in a lottery happening in a perfectly classical (I mean, not quantum) world. Then this process surely is deterministic; however, one usually model this situation with probabilities, saying something like "such ball has $0.02$ probability of being chosen".
