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Toby_Ord comments on A Request for Open Problems - Less Wrong
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Yes, and that's sort of intentional. I was trying to come up with a mathematical model of an agent that can deal with uncomputable physics. The physics of our universe seems likely to be computable, but there is no a priori reason to assume that it must be. We may eventually discover a law of physics that's not computable, or find out that we are in a simulation running inside a larger universe that has uncomputable physics. Agents using UTM-based priors can't deal with these scenarios.
So I tried to find a "better", i.e., more expressive, language for describing objects, but then realized that any fixed formal language has a similar problem. Here's my current idea for solving this: make the language extensible instead of fixed. That is, define a base language, and a procedure for extending the language. Then, when the agent encounters some object that can't be described concisely using his current language, he recursively extends it until a short description is possible. What the extension procedure should be is still unclear.
I agree that there are very interesting questions here. We have quite natural ways of describing uncomputable functions very far up the arithmetical hierarchy, and it seems that they can be described in some kind of recursive language even if the things they describe are not recursive (using recursive in the recursion theory sense both times). Turing tried something like this in Systems of Logic Based on Ordinals (Turing, 1939), but that was with formal logic and systems where you repeatedly add the Godel sentence of a system into the system as an axiom, repeating this into the transfinite. A similar thing could be done describing levels of computability transfinitely far up the arithmetical hierarchy using recursively represented ordinals to index them. However, then people like you and I will want to use certain well defined but non-recursive ordinals to do the indexing, and it seems to degenerate in the standard kind of way, just a lot further up the hierarchy than before.
And then there are objects that are completely outside the arithmetical hierarchy, but we probably shouldn't assign zero priors to either. Things like large cardinals, perhaps.
Another mystery is, why did evolution create minds capable of thinking about these issues, given that agents equipped with a fixed UTM-based prior would have done perfectly fine in our place, at least up to now?