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determinism

If you are allowed to think of a thing itself as its own model, and define determinism as ``modellable'' then everything is by definition deterministic. This just shows that one of these definitions is useless. What we want to say is ``modellable displaced in time'', but that requires a concept of time external to the system. To talk of a system without anything outside of it becomes impossible. (Kind of like the problem they have with the 2nd Law of Thermodynamics; we can't really prove it because there's no such thing as a closed system.)

I think perhaps it's impossible for us to prove that the universe is deterministic. Or at least meaningless, because we'd have no way of making use of that knowledge (no way of modelling the universe). Maybe God could, though, being part of the system outside the universe. But (pay attention here), the determinism of our universe is bound to the determinism of God's; i.e. if his is, so is ours and if his isn't, ours isn't. So to prove our universe is deterministic, he must first prove his own is, which we've seen is impossible. And this is true of a nested hierarchy of Gods as well. Which is just another way of saying that it's impossible to model anything, because if the model exists in the same system as the original, then they will affect each other!

Continous time implies determinism.

It only implies determinism if physical laws are immutable.

The short answer is that time doesn't exist if physical laws are mutable. The long answer:

Time is defined by motion, so if time is continous we're just saying all motion is continuous. In math terms, that means all motion can be described by functions differentiable on all orders. Or another way of thinking about it, is that you can't spontaneously "start" to move, or accellerate, or change accellerations; if you look closely enough, you should always find some instant when you were between changes. This is just following our intuition that there are limits to how things can change; I can't spontaneously appear on the other side of the room; I can't change speeds without accellerating; I can't change accellerations without "jerking" (I learned "jerk" is the word for d^3s/dt^3).

All these together imply differentiability. If a path is smooth and differentiable, it can by definition be described by a function, and that's our physical law. In intuitive terms, we're saying everything is connected, everything flows from one state to the next.

You might say, what if there are only a few discontinuities, how would this disrupt the notion of time? Well, we could be talking about only a small discontinuity, like gravity not being obeyed for half a second, but it's all the same if we imagine a bigger one, like suddenly the whole universe changed to a different universe.

Would it be possible, after the "change", to know it had happened? No. The whole principle of time is that events can be ordered and the separation between them can be measured. This is possible because motion is smooth; it allows us to relate and order events because they are connected by smooth motion. There's no way to describe time when the whole universe suddenly changes because there is no connection between events; what happened before is suddenly unrelated and time is meaningless.

That's the big step, to see that time is defined by motion. Time is not some independent thing ticking away; we only have time because the "smooth" nature of motion allows events to be ordered.

Of course, that may indeed happen. That is we may indeed be entering a new universe every second and not know it. But this is metaphysical nonsense on the order of "what if we're just old guys hooked up to computers"; such a situation is completely indistinguishable from reality, so to worry about it is a waste of time and we just apply Occam's razor.

If a small part of the universe were non-deterministic, then the resulting non-determinism would tend to "leak out". I.e. nothing is really an isolated system.