Question

Hi, a short while ago I finished reading your book The Science of Freedom and enjoyed it very much.
But, like a good Pole, there are a few points I didn’t agree with. And I’d be happy to hear your opinion—mainly about Newcomb’s experiment (pages 150–154).
In the book it is presented as proof that determinism cannot be true, or at least not determinism together with free choice. Because basically, if there is a prophet who already knows what you will do, then you have no free choice. (Since your future is determined by his prophecy.) And you broaden the argument and say that because there is no correct choice for a person with a “will,” a deterministic picture also cannot hold. But my claim is relevant even without that extension.
I want to disagree and say that nothing at all about free choice can be learned from Newcomb’s experiment (or even about determinism). Because the existence of the prophet is self-contradictory due to Turing’s halting problem. That’s all. But to get to the precise issue, I’ll elaborate:
After all, in mathematics and logic we often use non-existent “monsters” in order to learn about what does exist. For example, we say many things about a nondeterministic Turing machine even though in our world it does not exist. (Although it can be simulated but then the runtime grows exponentially, and therefore a “real” machine like that, with all its benefits, does not actually exist.)
But all that is only as long as there is no internal contradiction in the concept itself, however far it may be from reality. 
To reach the contradiction, let’s make 2 moves:
1] In our case there are 2 human beings, one chooser and one prophet. And in my view, already here there is a point that needs correction. Because if, say, I brought a cricket that chooses based on day/night, then there is no paradox here at all, and the immediate claim would be that I brought someone too stupid as the chooser, without the capacity to be a vessel for free choice—and that says nothing about everyone else. If so, why should the chooser be a human being? The conclusion is supposed to be about determinism in general. So why wouldn’t it be required that the chooser be as intelligent as possible for the paradox to hold, assuming it can hold in a deterministic world? If so, let’s replace the human being with a supercomputer with at least as many processors as the number of human neurons (not practical nowadays, of course).
2] But—and here comes the second stage—because of Turing’s “halting problem,” which explicitly applies to a supercomputer too, however powerful it may be, we know that there is no program that can determine for *every* other program whether it will halt. And following that there is Rice’s theorem, which says that any non-trivial property as well (like which box, for example) cannot be decided for *every* program.
And if your response is to bring a program that is simple enough, or to add a rule for the case such as a time limit, then again we return to the first move, because you have shown that that specific situation cannot support free choice—but who says that a more general situation could not support free choice, and how could you infer globally from that?
So in fact my claim is: from a certain level of automaton upward (a Turing machine), there is no longer any prophet, and therefore the experiment cannot take place as Newcomb describes it. So no conclusion about determinism and free choice—at least from this experiment—follows at all (either for or against).

Later in the chapter, you basically try to answer the contradiction and say that because “the information exists,” for example in brain scanners, therefore a prophet also exists in that deterministic world. But the “halting problem” actually says the opposite. There is information that you will never know how to get to in advance. Not from brain scanners and not from the electricity in the processor. Because the problem lies in predicting the algorithm itself, which is fully exposed to everyone. 

And a few more comments:

• Human beings are like Turing machines, but they have drawbacks such as fatigue, and perhaps brain waves that can be predicted because of their inferiority, so I replaced them with a processor in order to reach the contradiction faster
• The parable itself can still be useful for other conclusions, such as in game theory, since the conclusion there affects how a person should choose, and not whether fundamentally he has choice, because the prophet there could have a time machine. Which is not relevant here.
• Interestingly, no one addressed the problem I’m raising, and even Wikipedia ignored it despite discussing free will in light of Newcomb’s experiment here. But I still found 2 people on the internet (here and here) who raised the same question I did… and at least there there were no answers that solved the difficulty, so I have some confidence that I’m not talking nonsense and wasting your time

What do you think? 

Thanks in advance, Yoni.