Regarding Newcomb’s experiment described in the book The Science of Freedom

Hi, I recently finished reading your book “The Science of Freedom” and I really enjoyed it.
But, like a good Pole, there are some points I disagreed with. And I would love to hear your opinion. Especially about Newcomb’s experiment (page 150-154)
In the book it is presented as evidence that determinism is not possible, or at least determinism together with free choice. Because basically because there is a prophet who already knows what you will do, you do not have free choice. (Since your future is determined by his prophecy) and you expand the scope and say that because there is no right choice for a person with “will”, then a deterministic picture cannot exist either. But my argument is relevant even without the expansion.
I would like to disagree and say that from Newcomb’s experiment, no one can learn anything about free choice (or even about determinism), because the existence of the Prophet is a contradiction in terms due to Turing’s halting problem. That’s all. But to get to the exact problem, I will expand:
Because in mathematics and logic, we often use non-existent “monsters” to learn about what exists. For example, we say a lot about non-deterministic Turing machines even though they don’t exist in our world. (Although it can be symbolized , but then the runtime increases exponentially , and therefore such a “real” machine with all its benefits really doesn’t exist.)
But all this as long as there is no internal contradiction in the concept itself, no matter how far from reality it may be.
To reach a contradiction, we make 2 moves:
1] In our case, there are 2 people, one chooses and one predicts. And in my opinion, there is already a point for correction here. Because if we were to suppose that I were to bring a cricket that would choose according to day/night, then there is no paradox here at all and the immediate claim is that I brought someone too stupid as the chooser without the possibility of being a tool of capacity for free choice, and that does not say anything about the rest. And if so, why would the person choose? After all, the conclusion is about determinism as a whole. And if so, why is it not required that the chooser be as smart as possible so that the paradox can exist as long as it can exist in a deterministic world? If so, we would replace the person with a supercomputer with at least as many processors as the number of neurons in the person (impractical in our day, of course)
2] But, and here comes the second stage, because due to Turing’s “halting problem”, which explicitly applies to a supercomputer, no matter how powerful, we know that there is no program that knows for * any * other program whether it will halt. And further to that is Rice’s theorem , which claims that even any non-trivial property (like a box, for example) cannot be determined for * any * program.
And if you respond by bringing in a simple enough program, or by adding a rule for a case like a time limit, then we will be back to the first move, since you showed that that situation cannot accept free choice, but who said that a more general situation cannot accept free choice, and how would you conclude globally?
And in fact, I argued, from a certain level of automaton (known as Turing), there is no longer a prophet, and therefore the experiment cannot take place as Newcombe describes. Therefore, any conclusion about determinism and free choice, at least from this experiment, does not emerge (for or against).

Later in the chapter, you actually try to answer the contradiction and say that because “the information exists” from, for example, brain waves, therefore a prophet also exists in that deterministic world, but the “halting problem” actually claims the opposite. There is information that you will never know how to reach in advance. Neither from brain waves nor from the electricity in the processor. Because the problem is in predicting the algorithm itself, which is visible to everyone.

And more notes:

• Humans are like Turing machines, but they have drawbacks like fatigue, and maybe brain waves that predict their inferiority, so I replaced them with a superprocessor to get to the contradiction faster.
• The parable itself can still be used for other conclusions, such as in game theory, since the conclusion there reflects on how a person should choose, and not whether he fundamentally has a choice, since the prophet there could have a time machine. Which is irrelevant here.
• Interestingly, no one addressed the problem I raise, and even Wikipedia ignored it despite talking about free choice in the Newcomb experiment here , but I still found 2 people online ( here and here ) who raised the same question as me… and at least there were no answers there that solved the problem, so I have some confidence that I’m not talking nonsense and wasting your time.

What do you think?

Thanks in advance, June.