Self-referential systems cannot predict their future behaviour? (reference request)

Question

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?

Answer 1

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).