Try considering a Turing Machine which is running several Turing Machines within it. Call it A. Within it are B and C. B gets an input state that is the entire Turing Machine. Meanwhile C gets the output of B. There are multiple B and multiple BC. So we get something like this when we write it out:
A = B + B + B + BC + BC + BC
You might be tempted to say that A = A, therefore A is deterministic. This would even be logically true. However, it is actually far more dangerous than it appears. Why? Well, you aren't A. You are within A. Lets say you are B within A. Does it matter that you know the dynamics function? It is deterministic, but does it follow from it being deterministic that it is deterministic? No! Even though B is within the deterministic system A, B cannot claim that A is deterministic, because B cannot claim that C is deterministic, because C is indeterminable by B through self-reference.
Now the standard mistake is to let B run then let C run then to pretend B could determine C, because now obviously we can tell that A was determined by B because C was determined by B. Which sounds really compelling, but ask yourself this to see why it isn't as great as you think it is: is the universe still subject to a deterministic dynamics function right now? If so, can you tell me what C is, not in the toy problem, but in the real world?
Like, I realize this is an impossible problem: I'm asking you for the current configuration over the course of the next second for all matter in our universe even the parts you can't model because you haven't observed it. But that is kind of my point. The universe hasn't stopped running yet. You can't determine C from the information context of B.
Now lets say you try to counter, ah, but A so therefore all the other stuff is meaningless.
It seems like it works, but lets talk about the parable of the time you were given a deterministic dynamics function and you computed it according to what it was because you wanted to claim it was deterministic by treating it as what it was.
So you start calculating B + B + B + BC + BC + BC. And you see me looking at the same problem and idiotic like I am, you see me write down A = A' = BC. Then I get ready to solve it. And you are like, pfft, what an idiot. He isn't even talking about, like, actual reality.
unfortunately, in the calculations that follow, you realize something strange: Though you may B, I C you. And because I see you before you be such that I can determine what C should be such that you cannot see C while you B be, I choose a C for your B so that your B doesn't see C.
Basically, in choosing the slowest path, you also choose to be determinable, but in choosing being determinable, you implicitly make yourself vulnerable to an A' that makes your context - during the process - undecidable.
You thought choosing A saved you, but actually it was your curse:
You let me set A = B + B + B + A' because you didn't want to claim that we could divide A into the pieces which it was in actuality composed. But because it was in actuality composed of those pieces BC is now calculatable by A' such that your conceit is actually what traps you in the non-deterministic perspective. If B was calculated, not via A's conception of B, but A' then now it isn't just B that can't predict C. C probably can't predict B either.
So lets escape this! Lets say the process ends! Now B has fully determined itself such that C is determined and now A is determined as well.
Is A now deterministic? No, A isn't anything anymore.
Nothing is happening.
A isn't now deterministic; A was determined and now our physics is stopped. Where is the determinism at? Not within A, because clearly during A B wasn't deterministic because of self-reference to C. Not afterward either, because now it isn't doing anything.
Try considering a Turing Machine which is running several Turing Machines within it. Call it A. Within it are B and C. B gets an input state that is the entire Turing Machine. Meanwhile C gets the output of B. There are multiple B and multiple BC. So we get something like this when we write it out:
A = B + B + B + BC + BC + BC
You might be tempted to say that A = A, therefore A is deterministic. This would even be logically true. However, it is actually far more dangerous than it appears. Why? Well, you aren't A. You are within A. Lets say you are B within A. Does it matter that you know the dynamics function? It is deterministic, but does it follow from it being deterministic that it is deterministic? No! Even though B is within the deterministic system A, B cannot claim that A is deterministic, because B cannot claim that C is deterministic, because C is indeterminable by B through self-reference.
Now the standard mistake is to let B run then let C run then to pretend B could determine C, because now obviously we can tell that A was determined by B because C was determined by B. Which sounds really compelling, but ask yourself this to see why it isn't as great as you think it is: is the universe still subject to a deterministic dynamics function right now? If so, can you tell me what C is, not in the toy problem, but in the real world?
Like, I realize this is an impossible problem: I'm asking you for the current configuration over the course of the next second for all matter in our universe even the parts you can't model because you haven't observed it. But that is kind of my point. The universe hasn't stopped running yet. You can't determine C from the information context of B.