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Seth Lloyd claims in this video that free will stems from our impossibility of determining how a system capable of self-reference will behave in the future (e.g. whether a human being will have chosen to drink tea or coffee in 5 minutes). He relates this to Gödel's incompleteness theorem in a way that seems completely unjustified, but perhaps I misunderstand his point. Does anyone have a more rigorous reference to the claim that self-referential systems cannot predict their own future behaviour?

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  • $\begingroup$ I think you'll find most armchair philosophers fall down in presupposing that "free will" exists and that we all already know what it means. They may as well be talking about how the soul "stems from our impossibility etc.". Whilst "free will" is not overtly supernatural, it is essentially a foe of scientific enquiry on par with supernatural belief, in that it posits the existence of a "will"... (1/2) $\endgroup$
    – Steve
    Jan 10, 2023 at 19:44
  • $\begingroup$ ... that governs the individual free of any natural law (by analogy with the will of a deity that governs the natural world free of any natural law). In this respect it conflicts with the defining scientific axiom, that nature is deterministic. (2/2) $\endgroup$
    – Steve
    Jan 10, 2023 at 19:44

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I'm not going to watch or engage with the video but to answer your question, it is possible for a self-referencing program to make statements about its own behaviour.

Fix an encoding of programs. There is a program with index $e$ that takes another program index as an argument and searches for proofs about the behaviour of the program defined by this index.

Apply the $S^m_n$ theorem to show that there is a computable-function $s^0_1$ such that $\varphi_{s^0_1(e, n)} \cong \varphi_e(n)$. This means that when given a program index, we can construct a new program that takes no input and searches for proofs of the behaviour of the program with the index specified.

The recursion theorem tells us that $s^0_1$ has a fixed point. So there is some $n$ such that $\varphi_n \cong \varphi_{s^0_1(e, n)}$. This means there is a program with index $n$ that computes the same function as a program searching for statements about its own behaviour.

If you're interested, see What are some interesting applications/corollaries of Kleene's Recursion theorem? for more implications of the recursion theorem.

[1]: https://mathoverflow.net/a/437912. Joel David Hamkins (accessed 2023-01-5).

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