"So am I right to equate the question of whether some events can occur or not, to the question of whether for all valid formulae in that system, there is a mechanical sequence of proof that infers the asserted formulae from the axioms of the system in finite number of steps? If that is the case, for an undecidable system, the answer is no, by definition.
I am not sure if that is what you meant though, and I have some trouble connecting the dots between determinism and decidability in a logical sense."
That’s approximately what I meant, although I was really thinking of undecidable systems as a kind of metaphor for addressing the problem (which is why it isn’t that easy to connect the dots). In my initial post I emphasized the importance of precisely defining one’s terms before getting embroiled in debates. The issue of “determinism” is an excellent example. The very word seems to mean different things to different people. That’s why we hear talk about “strong” vs. “weak determinism.” We also hear it said that “determinism” does not necessarily imply “fatalism,” although the latter would presumably imply the former. If that’s the case, then saying that an event “must” happen doesn’t necessarily mean that it was predictable at all. But in that case, what does it mean?
One answer that is sometimes given is that the event must have some prior cause. But is an event caused by a previous event, or is it caused by an entity? As an Aristelian-oriented thinker, I lean toward the latter view myself. From that viewpoint, however, determinism wouldn’t exclude free will, since the entity that caused the event might itself have the attribute of free will. And if free will isn’t excluded, there doesn’t seem to be much left in determinism at all. BTW, I was rereading some Aristotle over the holidays, and I was gratified to see him spell out an important distinction: Propositions regarding future events must necessarily be true or false, but that doesn’t necessarily imply that that when said events occur, they will have been determinied.
My specific thinking regarding Gödel’s work was that it provides us a way of conceptualizing how a nondeterministic universe might work. Consider the set of all propositions of arithmetic. (Naturally, I’m thinking here of “arithmetic” as it is understood by mathematicians, including not only the simple equations of elementary-school texts but also statements such as Goldbach’s conjecture (if it’s true), rendered into appropriately rigorous symbolic notation.) As you know, Gödel showed that in such a system not all propositions can be decided as true/false in a finite number of steps from a finite number of internally consistent axioms. OTOH, most mathematicians would argue (although there may admittedly be some intuition involved here) that the law of the excluded middle does apply to arithmetic propositions: that is, each one really is either true or false.
Now suppose that some physicist in the post-Singularity future was able to establish a one-to-one correspondence between conceivable events in the physical universe and the propositions of arithmetic (or perhaps a subset of the latter), such that each conceivable event actually occurs in the universe if and only if the corresponding proposition is true. Therefore, if we grant that the law of the excluded middle applies to arithmetic propositions, then it also applies to physical events, meaning that each conceivable event either will occur or it won’t. But at the same time, we could say that the events that do occur are not determined in those cases where they map into an undecidable (in the Gödelian sense) proposition.
As I said, this should only be taken as a metaphor, not as an assertion that reality actually works this way.
